Get It Done With MySQL 5&Up, Chapter 20. Copyright © Peter Brawley and Arthur Fuller 20052016. All rights reserved.
Graphs and SQL
Edge list
Edge list tree
CTE edge list treewalk
Automate tree drawing
Nested sets model of a tree
Edgelist model of a network
Parts explosions
Most nontrivial data is hierarchical. Customers have orders, which have line items, which refer to products, which have prices. Population samples have subjects, who take tests, which give results, which have subresults and norms. Web sites have pages, which have links, which collect hits across dates and times. These are hierarchies of tables. The number of tables limits the number of JOINs needed to walk the tree. For such data, standard SQL is excellent.
But when data maps a family tree, or a browsing history, or a bill of materials, data rows relate hierarchically to other rows in the same table. We no longer know how many JOINs we need to walk the tree. We need a different data model.
That model is the graph (Fig 1), which is a set of nodes (vertices) and the edges (lines or arcs) that connect them. This chapter is about how to model and query graphs in a MySQL database.
Graph theory is a branch of topology. It is the study of geometric relations that aren't changed by stretching and compression—rubber sheet geometry, some call it. Graph theory is ideal for modelling hierarchies—like family trees, browsing histories, search trees, Bayesian networks and bills of materials— whose shape and size we can't know in advance.
Let the set of nodes in Fig 1 be N
,
the set of edges be L
, and the graph be
G
. Then G
is the tuple or ordered pair {N,L}
:
N = {A,B,C,D,E,F}
L = {AC,CD,CF,BE}
G = {N,L}
If the edges are directed, the graph is a digraph or directed graph. A mixed graph has both directed and undirected edges.
Examples of graphs are organisational charts; itineraries; route maps; parts explosions; massively multiplayer games; language rules; chat histories; network and link analysis in a wide variety of fields, for example search engines, forensics, epidemiology and telecommunications; data mining; models of chemical structure hierarchies; and biochemical processes.
Nodes and edges : Two nodes are adjacent if there is an edge between them. Two edges are adjacent if they connect to a common node. In a complete graph, all nodes are adjacent to all other nodes.
In a digraph, the number of edges entering a node is its indegree; the number leaving is its outdegree. A node of indegree zero is a root node, a node of outdegree zero is a leaf node.
In a weighted graph, used for example to solve the
travelling salesman problem, edges have a weight attribute. A digraph
with weighted edges is a network.
Paths and
cycles: A connected sequence of edges is a path, its
length the number of edges traversed. Two nodes are connected if
there is a path between them. If there is a path connecting every
pair of nodes, the graph is a connected graph.
A path in which no node repeats is a simple path. A path which returns to its own origin without crossing itself is a cycle or circuit. A graph with multiple paths between at least one pair of nodes is reconvergent. A reconvergent graph may be cyclic or acyclic. A unit length cycle is a loop.
If a graph's edges intersect only at nodes, it is planar. Two paths having no node in common are independent.
Traversing graphs: There are two main approaches, breadthfirst and depthfirst. Breadthfirst traversal visits all a node's siblings before moving on to the next level, and typically uses a queue. Depthfirst traversal follows edges down to leaves and back before proceeding to siblings, and typically uses a stack.
Sparsity: A graph where the size of E approaches
the maximum N^{2} is dense. When the multiple is
much smaller than N, the graph is considered sparse.
Trees: A tree is a connected graph with no
cycles. It is also a graph where the indegree of the root node is 0,
and the indegree of every other node is 1. A tree where every node is
of outdegree <=2 is a binary tree. A forest
is a graph in which every connected component is a tree.
Euler
paths: A path which traverses every edge in a graph exactly once
is an Euler path. An Euler path which is a circuit is an Euler
circuit.
If and only if every node of a connected
graph has even degree, it has an Euler circuit (which is why
the good people of Königsberg cannot go for a walk crossing each
of their seven bridges exactly once). If and only if a connected
graph has exactly 2 nodes with odd degree, it has a noncircuit
Euler path. The degree of an endpoint of a noncycle Euler path
is 1 + twice the number of times the path passes through that node,
so it is always odd.
Traditionally, computer science textbooks have offered edge lists, adjacency lists and adjacency matrices as data structures for graphs, with algorithms implemented in languages like C, C++ and Java. More recently other models and tools have been suggested, including query languages customised for graphs.
Edge list: The simplest way to represent a graph
is to list its edges: for Fig 1, the edge list is
{AC,CD,CF,BE}
. It is easy to add an edge to the list; deletion
is a little harder.
Table 1 

Nodes 
Adjacent nodes 
A 
C 
B 
E 
C 
F,D,A 
D 
C 
E 
B 
F 
C 
Adjacency
list:
An adjacency
list is
a ragged array: for each node it lists all adjacent nodes. Thus it
represents a directed graph of n
nodes
as a list of n
lists
where list i
contains
node j
if
the graph has an edge from node i
to
node j.
An undirected graph may be represented by having node j
in
the list for node i,
and node i
in
the list for node j.
Table 1 shows the adjacency list of the graph in Fig 1 interpreted as
undirected.
Adjacency
matrix:
An adjacency
matrix represents
a graph with n
nodes
as an n
x
n
matrix,
where the entry at (i,j)
is
1 if there is an edge from node i
to
node j,
or zero if there is not.
An adjacency matrix can represent a weighted graph using the weight as the entry, and can represent an undirected graph by using the same entry in both (i,j) and (j,i), or by using an upper triangular matrix.
There are useful glossaries here and here.
Standard SQL has been cumbersome for the recursive rowtorow logic of graphs. To fix this, DB2, Oracle, SQL Server and PostgreSQL have added recursive Common Table Expressions (CTE s). Before 8.0, MySQL did not have CTEs, so recursive graph traversal required stored routines. MariaDB has had CTEs since version 10.2.2 , MySQL since version 8.0.2. Joe Celko and Scott Stephens, among others, have published general SQL graph problem solutions that are simpler and smaller than equivalent C++, C# or Java code. Here we show how to use such tools.
Beware that in ports of edge list and adjacency list methods to SQL, there has been name slippage. What SQLers often call an adjacency list isn't like the adjacency list shown in Table 1; it's an edge list. Here we'll honour that fact, and mostly call them edge lists, but to keep the peace we’ll sometimes call them edgeadjacency lists.
There are also two newer kinds of models: what Joe Celko calls the nested sets model—an interval model using greaterthan/lessthan arithmetic to encode tree relationships and modified preorder tree traversal (MPTT) to query them, and Tropashko's materialised path model, where each node is stored with its (denormalised) path to the root. So now we have five main ways to model graphs in MySQL:
edgeadjacency lists: based on an adaptation by EF Codd of the logic of linked lists to table structures and queries,
adjacency matrices,
nested sets for trees simplify some queries, but insertion and deletion are cumbersome, and
materialised paths,
recursive CTEs.
The edge list is the simplest possible SQL representation of a graph: minimally, a single edges table where each row specifies one node and its parent (which is NULL for the root node), or to avoid DKNF problems, two tables: one for the nodes, the other to link the edges.
In the real world, the nodes table might be a table of personnel, or assembly parts, or locations on a map. It might have many other columms of data. The edges table might also have additional columns for edge properties. The key integers of both tables might be BIGINTs.
To model Fig 1, though, we keep things as simple as possible:
Listing 1
CREATE TABLE nodes(
nodeID CHAR(1) PRIMARY KEY
);
CREATE TABLE edges(
childID CHAR(1) NOT NULL,
parentID CHAR(1) NOT NULL,
PRIMARY KEY(childID,parentID)
);
INSERT INTO nodes VALUES('A'), ('B'), ('C'), ('D'), ('E'), ('F');
INSERT INTO edges VALUES ('A','C'), ('C','D'), ('C','F'), ('B','E');
SELECT * FROM edges;
+++
 childID  parentID 
+++
 A  C 
 B  E 
 C  D 
 C  F 
+++
Now, without any assumptions about whether the graph is connected, whether it is directed, whether it is a tree, or whatever, how hard is it to write a reachability procedure, a procedure which tells us where we can get to from here, wherever 'here' is?
A simple approach is a breadthfirst search:
Seed the list with the starting node,
Add, but do not duplicate, nodes which are children of nodes in the list,
Add, but do not duplicate, nodes which are parents of nodes in the list,
Repeat steps 2 and 3 until there are no more nodes to add.
Here it is as a MySQL stored procedure. It avoids duplicate nodes
by defining reached.nodeID
as a primary
key and adding reachable nodes with INSERT
IGNORE
:
Listing 2
DROP PROCEDURE IF EXISTS ListReached;
DELIMITER go
CREATE PROCEDURE ListReached( IN root CHAR(1) )
BEGIN
DECLARE rows SMALLINT DEFAULT 0;
DROP TABLE IF EXISTS reached;
CREATE TABLE reached (
nodeID CHAR(1) PRIMARY KEY
) ENGINE=HEAP;
INSERT INTO reached VALUES (root );
SET rows = ROW_COUNT();
WHILE rows > 0 DO
INSERT IGNORE INTO reached
SELECT DISTINCT childID
FROM edges AS e
INNER JOIN reached AS p ON e.parentID = p.nodeID;
SET rows = ROW_COUNT();
INSERT IGNORE INTO reached
SELECT DISTINCT parentID
FROM edges AS e
INNER JOIN reached AS p ON e.childID = p.nodeID;
SET rows = rows + ROW_COUNT();
END WHILE;
SELECT * FROM reached;
DROP TABLE reached;
END;
go
DELIMITER ;
CALL ListReached('A');
++
 nodeID 
++
 A 
 C 
 D 
 F 
++
To make the procedure more versatile, give it input parameters to tell it whether to list child, parent or all connections, and whether to recognise loops (for example C to C).
To give the model referential integrity, use InnoDB and make
edges.childID
and edges.parentID
foreign keys. To add or delete a node, add or delete desired single
rows in nodes
and edges
.
To change an edge, edit it. The model does not require the graph to
be connected or treelike, and does not presume direction.
The edge list is basic to what SQLers often call the adjacency list model.
Writers in the SQL graph literature often give solutions using single denormalised tables. Denormalisation can cost, big time. The bigger the table, the bigger the cost. You cannot edit nodes and edges separately. Carrying extra node information during edge computation slows performance.
To avoid such difficulties, normalise trees like William Shakespeare's family tree (Fig 2) into two tables, nodes (family) with a row for each individual's information, about individuals, and edges (familytree)with a row for each parentchild link or edge.
Listing 3
 Base data:
CREATE TABLE family (
ID smallint unsigned PRIMARY KEY AUTO_INCREMENT,
name char(20) default '',
siborder tinyint(4) default NULL,
born smallint(4) unsigned default NULL,
died smallint(4) unsigned default NULL
);
INSERT INTO family VALUES (1, 'Richard Shakespeare', NULL, NULL, 1561);
INSERT INTO family VALUES (2, 'Henry Shakespeare', 1, NULL, 1569);
INSERT INTO family VALUES (3, 'John Shakespeare', 2, 1530, 1601);
INSERT INTO family VALUES (4, 'Joan Shakespeare', 1, 1558, NULL);
INSERT INTO family VALUES (5, 'Margaret Shakespeare', 2, 1562, 1563);
INSERT INTO family VALUES (6, 'William Shakespeare', 3, 1564, 1616);
INSERT INTO family VALUES (7, 'Gilbert Shakespeare', 4, 1566, 1612);
INSERT INTO family VALUES (8, 'Joan Shakespeare', 5, 1568, 1646);
INSERT INTO family VALUES (9, 'Anne Shakespeare', 6, 1571, 1579);
INSERT INTO family VALUES (10, 'Richard Shakespeare', 7, 1574, 1613);
INSERT INTO family VALUES (11, 'Edmond Shakespeare', 8, 1580, 1607);
INSERT INTO family VALUES (12, 'Susana Shakespeare', 1, 1583, 1649);
INSERT INTO family VALUES (13, 'Hamnet Shakespeare', 1, 1585, 1596);
INSERT INTO family VALUES (14, 'Judith Shakespeare', 1, 1585, 1662);
INSERT INTO family VALUES (15, 'William Hart', 1, 1600, 1639);
INSERT INTO family VALUES (16, 'Mary Hart', 2, 1603, 1607);
INSERT INTO family VALUES (17, 'Thomas Hart', 3, 1605, 1670);
INSERT INTO family VALUES (18, 'Michael Hart', 1, 1608, 1618);
INSERT INTO family VALUES (19, 'Elizabeth Hall', 1, 1608, 1670);
INSERT INTO family VALUES (20, 'Shakespeare Quiney', 1, 1616, 1617);
INSERT INTO family VALUES (21, 'Richard Quiney', 2, 1618, 1639);
INSERT INTO family VALUES (22, 'Thomas Quiney', 3, 1620, 1639);
INSERT INTO family VALUES (23, 'John Bernard', 1, NULL, 1674);
 Table which models the tree:
CREATE TABLE familytree (
childID smallint unsigned NOT NULL,
parentID smallint unsigned NULL,
PRIMARY KEY (childID, parentID);
);
INSERT INTO familytree VALUES
(2, 1), (3, 1), (4, 2), (5, 2), (6, 2), (7, 2), (8, 2), (9, 2),
(10, 2), (11, 2), (12, 6), (13, 6), (14, 6), (15, 8), (16, 8),
(17, 8), (18, 8), (19, 12), (20, 14), (21, 14), (22, 14), (23, 19);
(The family
PK is autoincrement, but
the listing is more readerfriendly when the ID
values are shown.)
It will be useful to have a function that returns family.name
for a parent or child ID in familytree
:
Listing 4
DROP FUNCTION IF EXISTS PersonName; CREATE FUNCTION PersonName(pid smallint) RETURNS VARCHAR(20) DETERMINISTIC RETURN (SELECT name FROM family WHERE ID=pid);SELECT PersonName( parentID ) AS 'Father of William'
FROM familytree
WHERE childID = 6;
++
 Father of William 
++
 Henry Shakespeare 
++
SELECT PersonName( childID ) AS 'Children of William'
FROM familytree
WHERE parentID = ( SELECT ID FROM family WHERE name = 'William Shakespeare' );
++
 Children of William 
++
 Susana Shakespeare 
 Hamnet Shakespeare 
 Judith Shakespeare 
++
SELECT PersonName(childID) AS child, PersonName(parentID) AS parent
FROM familytree;
+++
 child  parent 
+++
 Henry Shakespeare  Richard Shakespeare 
 John Shakespeare  Richard Shakespeare 
 Joan Shakespeare  Henry Shakespeare 
 Margaret Shakespeare  Henry Shakespeare 
 William Shakespeare  Henry Shakespeare 
 Gilbert Shakespeare  Henry Shakespeare 
 Joan Shakespeare  Henry Shakespeare 
 Anne Shakespeare  Henry Shakespeare 
 Richard Shakespeare  Henry Shakespeare 
 Edmond Shakespeare  Henry Shakespeare 
 Susana Shakespeare  William Shakespeare 
 Hamnet Shakespeare  William Shakespeare 
 Judith Shakespeare  William Shakespeare 
 William Hart  Joan Shakespeare 
 Mary Hart  Joan Shakespeare 
 Thomas Hart  Joan Shakespeare 
 Michael Hart  Joan Shakespeare 
 Elizabeth Hall  Susana Shakespeare 
 Shakespeare Quiney  Judith Shakespeare 
 Richard Quiney  Judith Shakespeare 
 Thomas Quiney  Judith Shakespeare 
 John Bernard  Elizabeth Hall 
+++
A sametable foreign key can simplify tree maintenance:
Listing 4a create table edges ( ID int PRIMARY KEY, parentid int, foreign key(parentID) references edges(ID) ON DELETE CASCADE ON UPDATE CASCADE ) engine=innodb; insert into edges(ID,parentID) values (1,null),(2,1),(3,1),(4,2); select * from edges; +++  ID  parentid  +++  1  NULL   2  1   3  1   4  2  +++ delete from edges where id=2; select * from edges; +++  ID  parentid  +++  1  NULL   3  1  +++
Simple queries retrieve basic facts about the tree, for example GROUP_CONCAT() collects parent nodes with their children in correct order:
Listing 5
SELECT parentID AS Parent, GROUP_CONCAT(childID ORDER BY siborder) AS Children
FROM familytree t
JOIN family f ON t.parentID=f.ID
GROUP BY parentID;
+++
 Parent  Children 
+++
 1  3,2 
 2  4,5,6,7,8,9,10,11 
 6  12,13,14 
 8  18,17,16,15 
 12  19 
 14  22,21,20 
 19  23 
+++
Iterate over those commaseparated lists with a bit of application code and you have a hybrid treewalk. The paterfamilias is the root node, individuals with no children are the leaf nodes, and queries to retrieve subtree statistics are straightforward:
Listing 6
SELECT
PersonName(ID) AS Paterfamilias,
IFNULL(born,'?') AS Born,
IFNULL(died,'?') AS Died
FROM family AS f
LEFT JOIN familytree AS t ON f.ID=t.childID
WHERE t.childID IS NULL;
++++
 Paterfamilias  Born  Died 
++++
 Richard Shakespeare  ?  1561 
++++
SELECT
PersonName(ID) AS Childless,
IFNULL(born,'?') AS Born,
IFNULL(died,'?') AS Died
FROM family AS f
LEFT JOIN familytree AS t ON f.ID=t.parentID
WHERE t.parentID IS NULL;
++++
 Childless  Born  Died 
++++
 John Shakespeare  1530  1601 
 Joan Shakespeare  1558  ? 
 Margaret Shakespeare  1562  1563 
 Gilbert Shakespeare  1566  1612 
 Anne Shakespeare  1571  1579 
 Richard Shakespeare  1574  1613 
 Edmond Shakespeare  1580  1607 
 Hamnet Shakespeare  1585  1596 
 William Hart  1600  1639 
 Mary Hart  1603  1607 
 Thomas Hart  1605  1670 
 Michael Hart  1608  1618 
 Shakespeare Quiney  1616  1617 
 Richard Quiney  1618  1639 
 Thomas Quiney  1620  1639 
 John Bernard  ?  1674 
++++
SELECT ROUND(AVG(diedborn),2) AS 'Longevity of the childless'
FROM family AS f
LEFT JOIN familytree AS t ON f.ID=t.parentID
WHERE t.parentID IS NULL;
++
 Longevity of the childless 
++
 25.86 
++
In striking contrast with Celko's nested
sets model, inserting a new item in this model requires no
revision of existing rows. We just add a new family
row, then a new familytree
row with IDs
specifying who is parent to whom. Deletion is also a twostep: delete
the familytree
row for that childparent
link, then delete the family
row for
that child.
Edge list tree traversal is supposed to be difficult. Usually we don't know in advance how many levels the tree has, so the query needs recursion or a logically equivalent loop. Without CTEs (i.e.,, before MySQL 8.0 or MariaDB 10.2.2), that requires a stored procedure. Here is a simple algorithm that just seeds a result table with children of the root node, then adds remaining edges with INSERT IGNORE:
Listing 7
DROP PROCEDURE IF EXISTS famsubtree;
DELIMITER go
CREATE PROCEDURE famsubtree( root INT )
BEGIN
DROP TABLE IF EXISTS famsubtree;
CREATE TABLE famsubtree( childID smallint unsigned not null,
parentID smallint unsigned null, Primary Key(childID,parentID) )
SELECT childID, parentID, 0 AS level
FROM familytree
WHERE parentID = root;
REPEAT
INSERT IGNORE INTO famsubtree
SELECT f.childID, f.parentID, s.level+1
FROM familytree AS f
JOIN famsubtree AS s ON f.parentID = s.childID;
UNTIL Row_Count() = 0 END REPEAT;
END ;
go
DELIMITER ;
call famsubtree(1);  from the root you can see forever
SELECT Concat(Space(level),parentID) AS Parent, Group_Concat(childID ORDER BY childID) AS Child
FROM famsubtree
GROUP BY parentID;
+++
 Parent  Child 
+++
 1  2,3 
 2  4,5,6,7,8,9,10,11 
 6  12,13,14 
 8  15,16,17,18 
 12  19 
 14  20,21,22 
 19  23 
+++
Simple and quick. The logic ports to any edge list. We can prove that right now by writing a generic version. GenericTree() just needs parameters for the name of the target table, the names of its child and parent ID columns, and the parent ID whose descendants are sought:
Listing 7a: Generalpurpose edge list tree walker DROP PROCEDURE IF EXISTS GenericTree; DELIMITER go CREATE PROCEDURE GenericTree( edgeTable CHAR(64), edgeIDcol CHAR(64), edgeParentIDcol CHAR(64), ancestorID INT ) BEGIN DECLARE r INT DEFAULT 0; DROP TABLE IF EXISTS subtree; SET @sql = Concat( 'CREATE TABLE subtree SELECT ', edgeIDcol,' AS childID, ', edgeParentIDcol, ' AS parentID,', '0 AS level FROM ', edgeTable, ' WHERE ', edgeParentIDcol, '=', ancestorID ); PREPARE stmt FROM @sql; EXECUTE stmt; DROP PREPARE stmt; ALTER TABLE subtree ADD PRIMARY KEY(childID,parentID); REPEAT SET @sql = Concat( 'INSERT IGNORE INTO subtree SELECT a.', edgeIDcol, ',a.',edgeparentIDcol, ',b.level+1 FROM ', edgeTable, ' AS a JOIN subtree AS b ON a.',edgeParentIDcol, '=b.childID' ); PREPARE stmt FROM @sql; EXECUTE stmt; SET r=Row_Count();  save row_count() result before DROP PREPARE loses the value DROP PREPARE stmt; UNTIL r < 1 END REPEAT; END ; go DELIMITER ;
To retrieve details like names and other data associated with node IDs, write a frontend query to join the subtree result table with the required detail table(s), for example:
CALL GenericTree('familytree','childID','parentID',1); SELECT Concat(Repeat( ' ', s.level), a.name ) AS Parent, b.name AS Child FROM subtree s JOIN family a ON s.parentID=a.ID JOIN family b ON s.childID=b.ID; +++  Parent  Child  +++  Richard Shakespeare  Henry Shakespeare   Richard Shakespeare  John Shakespeare   Henry Shakespeare  Joan Shakespeare   Henry Shakespeare  Margaret Shakespeare   Henry Shakespeare  William Shakespeare   Henry Shakespeare  Gilbert Shakespeare   Henry Shakespeare  Joan Shakespeare   Henry Shakespeare  Anne Shakespeare   Henry Shakespeare  Richard Shakespeare   Henry Shakespeare  Edmond Shakespeare   William Shakespeare  Susana Shakespeare   William Shakespeare  Hamnet Shakespeare   William Shakespeare  Judith Shakespeare   Joan Shakespeare  William Hart   Joan Shakespeare  Mary Hart   Joan Shakespeare  Thomas Hart   Joan Shakespeare  Michael Hart   Susana Shakespeare  Elizabeth Hall   Judith Shakespeare  Shakespeare Quiney   Judith Shakespeare  Richard Quiney   Judith Shakespeare  Thomas Quiney   Elizabeth Hall  John Bernard  +++
Is GenericTree() fast? On standard hardware it walks a 5,000node tree in less than 0.5 secs—much faster than a comparable nested sets query on the same tree! It has no serious scaling issues. And its logic can be used to prune: call GenericTree() then delete the listed rows. Better still, write a generic tree pruner from Listing 7a and a DELETE command. To insert a subtree, prepare a table of new rows, point its top edge at an existing node as parent, and INSERT it.
The edge list treewalk is logically recursive, so how about coding it recursively? Here is a recursive depthfirst PHP treewalk for the familytree and family tables:
Listing 7b: Recursive edge list subtree in PHP $info = recursivesubtree( 1, $a = array(), 0 ); foreach( $info as $row ) echo str_repeat( " ", 2*$row[4] ), ( $row[3] > 0 ) ? "<b>{$row[1]}</b>" : $row[1], "<br/>"; function recursivesubtree( $rootID, $a, $level ) { $childcountqry = "(SELECT COUNT(*) FROM familytree WHERE parentID=t.childID) AS childcount"; $qry = "SELECT t.childid,f.name,t.parentid,$childcountqry,$level " . "FROM familytree t JOIN family f ON t.childID=f.ID " . "WHERE parentid=$rootID ORDER BY childcount<>0,t.childID"; $res = mysql_query( $qry ); while( $row = mysql_fetch_row( $res )) { $a[] = $row; if( $row[3] > 0 ) $a = recursivesubtree( $row[0], $a, $level+1 ); // down before right } return $a; }
A query with a subquery, a fetch loop, and a recursive callthat's all there is to it. A nice feature of this algorithm is that it writes result rows in displayready order. To port this to MySQL, you must have set maximum recursion depth in my.cnf/ini or in your client:
Listing 7c: Recursive edge list subtree in MySQL SET @@SESSION.max_sp_recursion_depth=25; DROP PROCEDURE IF EXISTS recursivesubtree; DELIMITER go CREATE PROCEDURE recursivesubtree( iroot INT, ilevel INT ) BEGIN DECLARE irows,ichildid,iparentid,ichildcount,done INT DEFAULT 0; DECLARE cname VARCHAR(64); SET irows = ( SELECT COUNT(*) FROM familytree WHERE parentID=iroot ); IF ilevel = 0 THEN DROP TEMPORARY TABLE IF EXISTS _descendants; CREATE TEMPORARY TABLE _descendants ( childID INT, parentID INT, name VARCHAR(64), childcount INT, level INT ); END IF; IF irows > 0 THEN BEGIN DECLARE cur CURSOR FOR SELECT childid,parentid,f.name, (SELECT COUNT(*) FROM familytree WHERE parentID=t.childID) AS childcount FROM familytree t JOIN family f ON t.childID=f.ID WHERE parentid=iroot ORDER BY childcount<>0,t.childID; DECLARE CONTINUE HANDLER FOR SQLSTATE '02000' SET done = 1; OPEN cur; WHILE NOT done DO FETCH cur INTO ichildid,iparentid,cname,ichildcount; IF NOT done THEN INSERT INTO _descendants VALUES(ichildid,iparentid,cname,ichildcount,ilevel ); IF ichildcount > 0 THEN CALL recursivesubtree( ichildid, ilevel + 1 ); END IF; END IF; END WHILE; CLOSE cur; END; END IF; IF ilevel = 0 THEN  Show result table headed by name that corresponds to iroot: SET cname = (SELECT name FROM family WHERE ID=iroot); SET @sql = CONCAT('SELECT CONCAT(REPEAT(CHAR(32),2*level),IF(childcount,UPPER(name),name))', ' AS ', CHAR(39),'Descendants of ',cname,CHAR(39),' FROM _descendants'); PREPARE stmt FROM @sql; EXECUTE stmt; DROP PREPARE stmt; END IF; END; go DELIMITER ; CALL recursivesubtree(1,0); ++  Descendants of Richard Shakespeare  ++  HENRY SHAKESPEARE   Joan Shakespeare   Margaret Shakespeare   WILLIAM SHAKESPEARE   SUSANA SHAKESPEARE   ELIZABETH HALL   John Bernard   Hamnet Shakespeare   JUDITH SHAKESPEARE   Shakespeare Quiney   Richard Quiney   Thomas Quiney   Gilbert Shakespeare   JOAN SHAKESPEARE   William Hart   Mary Hart   Thomas Hart   Michael Hart   Anne Shakespeare   Richard Shakespeare   Edmond Shakespeare   John Shakespeare  ++
In MySQL this recursive treewalk can be up to 100 times slower than GenericTree(). Its slowness is comparable to that of a MySQL version of Kendall Willet's depthfirst edge list subtree algorithm:
Listing 7d: Depthfirst edge list subtree CREATE PROCEDURE depthfirstsubtree( iroot INT ) BEGIN DECLARE ilastvisited, inxt, ilastord INT; SET ilastvisited = iroot; SET ilastord = 1; DROP TABLE IF EXISTS descendants; CREATE TABLE descendants SELECT childID,parentID,1 AS ord FROM familytree; UPDATE descendants SET ord=1 WHERE childID=iroot; this: LOOP SET inxt = NULL; SELECT MIN(childID) INTO inxt FROM descendants  go down WHERE parentID = ilastvisited AND ord = 1 ; IF inxt IS NULL THEN  nothing down, so go right SELECT MIN(d2.childID) INTO inxt FROM descendants d1 JOIN descendants d2 ON d1.parentID = d2.parentID AND d1.childID < d2.childID WHERE d1.childID = ilastvisited; END IF; IF inxt IS NULL THEN  nothing right. so go up SELECT parentID INTO inxt FROM descendants WHERE childID = ilastvisited AND parentID IS NOT NULL; END IF; UPDATE descendants SET ord = ilastord + 1 WHERE childID = inxt AND ord = 1; IF ROW_COUNT() > 0 THEN SET ilastord = ilastord + 1; END IF; IF inxt IS NULL THEN LEAVE this; END IF; SET ilastvisited = inxt; END LOOP; END;
One reason Willet's is slower is that MySQL does not permit multiple references to a temporary table in a query. When all algorithms are denied temp tables, though, this algorithm is still slower than recursion, and both are much slower than GenericTree().
A simple procedure to retrieve a node's ancestors:
Listing 7e: Find node's ancestors CREATE PROCEDURE ancestors( pid int ) begin drop temporary table if exists _ancestors; create temporary table _ancestors(parent int); set @id = pid; repeat select parentID,count(*) into @parent,@y from familytree where childID=@id; if @y>0 then insert into _ancestors values(@parent); set @id=@parent; end if; until @parent is null or @y=0 end repeat; select * from _ancestors order by parent; end;
Finally, for MariaDB from 10.2.2 and MySQL from 8.0.1, here is a recursive treewalk using a CTE. Such queries (see SELECT / WITH in Chapter 6) have five parts: a WITH clause to declare the derived table; a query to initialise the derived table, in this case with the root node; a UNION command; the recursive join; and a final output SELECT. We built the familytree table without a row for the root, so the initialising SELECT creates it, but it needn't be shown in the result:
Listing 7f: Walk a tree with a CTE: WITH RECURSIVE treewalk AS ( SELECT CAST(1 AS UNSIGNED) AS childID,  UNION NEEDS EXACT TYPE MATCH CAST(NULL AS UNSIGNED) AS parentID, CAST(0 AS UNSIGNED) AS level, 0 AS siborder UNION ALL SELECT familytree.childID, familytree.parentID, treewalk.level+1 AS level, family.siborder FROM familytree JOIN treewalk ON familytree.parentID=treewalk.childID JOIN family ON family.ID=familytree.childID ) SELECT Concat( Space(level1), parentID ) AS Parent, level1 AS Depth, Group_Concat( childID ORDER BY siborder ) AS Children FROM treewalk WHERE level>0 GROUP BY treewalk.parentID ORDER BY treewalk.parentID;  Unset only_full_group_by sql_mode ++++  Parent  Depth  Children  ++++  1  0  2,3   2  1  4,5,6,7,8,9,10,11   6  2  13,12,14   8  2  18,15,16,17   12  3  19   14  3  20,21,22   19  4  23  ++++
The logic is that of Listing 7, but the CTE makes this treewalk about ten times faster. Note that
the query requires sql_mode
not to include only_full_group_by
.
If the graph being traversed is cyclic, avoid an endless loop by changing UNION ALL
to
UNION DISTINCT
.
Edge list tree queries perform faster, and are easier to write, than their reputation suggests—especially when CTEs are available. And edge tables are flexible. For a tree describing a parts explosion rather than a family, just add columns for weight, quantity, assembly time, cost, price and so on. Reports need only aggregate column values and sums. We'll revisit this near the end of the chapter.
Path enumeration in an edge list tree is almost as easy as depthfirst subtree traversal:
create a table for paths,
seed it with paths of unit length from the tree table,
iteratively add paths till there are no more to add.
MySQL's INSERT IGNORE
command simplifies the code by removing the need for a NOT
EXISTS(...) clause in the INSERT
... SELECT statement. Since adjacencies are logically
symmetrical, we make path direction the caller's choice, UP
or DOWN
:
Listing 8
DROP PROCEDURE IF EXISTS ListAdjacencyPaths;
DELIMITER go
CREATE PROCEDURE ListAdjacencyPaths( IN direction CHAR(5) )
BEGIN
DROP TABLE IF EXISTS paths;
CREATE TABLE paths(
start SMALLINT,
stop SMALLINT,
len SMALLINT,
PRIMARY KEY(start,stop)
) ENGINE=HEAP;
IF direction = 'UP' THEN
INSERT INTO paths
SELECT childID,parentID,1
FROM familytree;
ELSE
INSERT INTO paths
SELECT parentID,childID,1
FROM familytree;
END IF;
WHILE ROW_COUNT() > 0 DO
INSERT IGNORE INTO paths
SELECT DISTINCT
p1.start,
p2.stop,
p1.len + p2.len
FROM paths AS p1 INNER JOIN paths AS p2 ON p1.stop = p2.start;
END WHILE;
SELECT start, stop, len
FROM paths
ORDER BY start, stop;
DROP TABLE paths;
END;
go
DELIMITER ;
To find the paths from just one node, seed the paths
table with paths from the starting node, then iteratively search a
JOIN of familytree
and paths
for edges which will extend
existing paths in the userspecified direction:
Listing 8a
DROP PROCEDURE IF EXISTS ListAdjacencyPathsOfNode;
DELIMITER go
CREATE PROCEDURE ListAdjacencyPathsOfNode( IN node SMALLINT, IN direction CHAR(5) )
BEGIN
TRUNCATE paths;
IF direction = 'UP' THEN
INSERT INTO paths
SELECT childID,parentID,1
FROM familytree
WHERE childID = node;
ELSE
INSERT INTO paths
SELECT parentID,childID,1
FROM familytree
WHERE parentID = node;
END IF;
WHILE ROW_COUNT() > 0 DO
IF direction = 'UP' THEN
INSERT IGNORE INTO paths
SELECT DISTINCT
paths.start,
familytree.parentID,
paths.len + 1
FROM paths
INNER JOIN familytree ON paths.stop = familytree.childID;
ELSE
INSERT IGNORE INTO paths
SELECT DISTINCT
paths.start,
familytree.childID,
paths.len + 1
FROM paths
INNER JOIN familytree ON paths.stop = familytree.parentID;
END IF;
END WHILE;
SELECT start, stop, len
FROM paths
ORDER BY start, stop;
END;
go
DELIMITER ;
CALL ListAdjacencyPathsOfNode(1,'DOWN');
++++
 start  stop  len 
++++
 1  2  1 
 1  3  1 
 1  4  2 
 1  5  2 
 1  6  2 
 1  7  2 
 1  8  2 
 1  9  2 
 1  10  2 
 1  11  2 
 1  12  3 
 1  13  3 
 1  14  3 
 1  15  3 
 1  16  3 
 1  17  3 
 1  18  3 
 1  19  4 
 1  20  4 
 1  21  4 
 1  22  4 
 1  23  5 
++++
Listing 8b: List an individual's ancestors (path to root):
WITH RECURSIVE ctepath AS (
SELECT parentID FROM familytree WHERE childID=23  INDIVIDUAL’S FATHER
UNION ALL
SELECT f.parentID FROM familytree f  AND THAT INDIVIDUAL’S FATHER ETC
JOIN ctepath ON f.childID=ctepath.parentID
)
SELECT Group_Concat(parentID) As AncestorsOf23 FROM ctepath;  RETURNS 19,12,6,2,1
These algorithms don't bend the brain. They perform acceptably with large trees, faster with CTEs. Querying edgeadjacency lists for subtrees and paths is less daunting than their reputation suggests.
Tables of numbers may be the most boring objects on earth. How to bring them alive? The Google Visualization API library has an ‘OrgChart’ module that can make edge list trees look like Fig 2, but each instance needs fifty or so lines of specific JavaScript code, plus an additional line of code for each row of data in the tree. Could we autogenerate that code? Mais oui! The module needs child node and parent node columns of data, and accepts an optional third column for info that pops up when the mouse hovers. Here is such a query for the Shakespeare family tree ...
Listing 9a
select concat( node.ID,' ', node.name) as node,
if( edges.parentID is null, '', concat(parent.ID, ' ',parent.name)) as parent,
if( node.born is null, 'Birthdate unknown', concat( 'Born ', node.born )) as tooltip
from family as node
left join familytree as edges on node.ID=edges.childID
left join family as parent on edges.parentID=parent.ID;
and here is a PHP function to generate HTML and JavaScript that paints an OrgChart for any tree query returning string columns for node, parent and optionally tooltips:
Listing 9b
function orgchart( $conn, $qry ) {
$cols = array(); $rows = array();
$res = mysqli_query( $conn, $qry ) or exit( mysqli_error($conn) );
$colcount = mysqli_num_fields( $res );
if( $colcount < 2 ) exit( "Org chart needs two or three columns" );
$rowcount = mysqli_num_rows( $res );
for( $i=0; $i<$colcount; $i++ ) $cols[] = mysqli_fetch_field( $res, $i );
while( $row = mysqli_fetch_row($res) ) $rows[] = $row;
echo "<html>\n<head>\n",
" <script type='text/javascript' src='https://www.google.com/jsapi'></script>\n",
" <script type='text/javascript'>\n",
" google.load('visualization', '1', {'packages':['orgchart']});\n",
" google.setOnLoadCallback(drawChart);\n",
" function drawChart() {\n",
" var data = new google.visualization.DataTable();\n";
for( $i=0; $i<$colcount; $i++ ) echo " data.addColumn('string','{$cols[$i]>name}')\n";
echo " data.addRows([\n";
for( $j=0; $j<$rowcount; $j++ ) {
$row = $rows[$j];
$c = (( $j < $rowcount1 ) ? "," : "" );
echo " ['{$row[0]}','{$row[1]}','{$row[2]}']$c\n";
}
echo " ]);\n",
" var chart = new google.visualization.OrgChart(document.getElementById('chart_div'));\n",
" var options = {'size':'small','allowHtml':'true','allowCollapse':'true'};\n",
" chart.draw(data, options);\n",
" }\n",
" </script>\n/head>\n<body>\n",
" <div id='chart_div'></div>\n",
"</body>\n</html>";
}
Imagine an oval drawn round every leaf and every subtree in Fig 2, and a final oval round the entire tree. The tree is a set. Each subtree is a subset. That's the basic idea of the nested sets model.
The advantage of the nested sets model is that root, leaves, subtrees, levels, tree height, ancestors, descendants and paths can be retrieved without recursion or application language code. The disadvantages are:
initial setup of the tree table can be difficult,
queries for parents (immediate superiors) and children (immediate subordinates) are more complicated than with an edge list model,
insertion, updates and deletion are extremely cumbersome since they may require updates to much of the tree.
The nested sets model depends on using a modified preorder tree traversal (MPTT) depthfirst algorithm to assign each node left and right integers which define the node's tree position. All nodes of a subtree have
left values greater than the subtree parent's left value, and
right values smaller than that of the subtree parent's right value.
so queries for subtrees are dead simple. If the numbering scheme is integersequential as in Fig 3, the root node receives a left value of 1 and a right value equal to twice the item count.
To see how to code nested sets using MPTT, trace the ascending
integers in Fig 3, starting with 1 on
the left side of the root node (Richard
Shakespeare
). Following edges downward and leftward, the left
side of each box gets the next integer. When you reach a leaf (Joan
,
left=3), the right
side of that box gets the next integer (4).
If there is another node to the right on the same level, continue in
that direction; otherwise continue up the right side of the
subtree you just descended. When you arrive back at the root on the
right side, you're done. Down, right and up.
A serious problem with this scheme jumps out right away: after you've written the Fig 3 tree to a table, what if historians discover an older brother or sister of Henry and John? Every row in the tree table must be updated!
Celko and others have proposed alternative numbering schemes to get round this problem, but the logical difficulty remains: inserts and updates can invalidate many or all rows, and no SQL CHECK or CONSTRAINT can prevent it. The nested sets model is not good for trees which require frequent updates, and is pretty much unsupportable for large updatable trees that will be accessed by many concurrent users. But as we'll see in a moment, it can be very useful indeed for reporting a tree.
Obviously, numbering a tree by hand would be errorprone, seriously impractical for large trees, so it's usually best to code the tree initially as an edge list, then use a stored procedure to translate the edge list representation to nested sets. Celko 's depthfirst pushdown stack method will translate any edge list tree into a nested sets tree, though slowly:
Create a table nestedsettree
for the tree: node
, leftedge
,
rightedge
, and a stack pointer (top
),
Seed that table with the root node of the edge list tree,
Set a nextedge counter to 1 plus the left value of the root node, i.e. 2,
While that counter is less than the rightedge value of the root node ...
insert a row for this parent's smallest unwritten child, and drop down a level, or
if we're out of children, increment rightedge , write it to the current row, and back up a level.
This version has been improved to handle edge list trees with or without a row containing the root node and its NULL parent:
Listing 10
DROP PROCEDURE IF EXISTS EdgeListToNestedSet;
DELIMITER go
CREATE PROCEDURE EdgeListToNestedSet( edgeTable CHAR(64), idCol CHAR(64), parentCol CHAR(64) )
BEGIN
DECLARE maxrightedge, rows SMALLINT DEFAULT 0;
DECLARE trees, current SMALLINT DEFAULT 1;
DECLARE nextedge SMALLINT DEFAULT 2;
DECLARE msg CHAR(128);
 create working tree table as a copy of edgeTable
DROP TEMPORARY TABLE IF EXISTS tree;
CREATE TEMPORARY TABLE tree( childID INT, parentID INT );
SET @sql = CONCAT( 'INSERT INTO tree SELECT ', idCol, ',', parentCol, ' FROM ', edgeTable );
PREPARE stmt FROM @sql; EXECUTE stmt; DROP PREPARE stmt;
 initialise result table
DROP TABLE IF EXISTS nestedsettree;
CREATE TABLE nestedsettree (
top SMALLINT, nodeID SMALLINT, leftedge SMALLINT, rightedge SMALLINT,
KEY(nodeID,leftedge,rightedge)
) ENGINE=HEAP;
 root is child with NULL parent or parent which is not a child
SET @nulls = ( SELECT Count(*) FROM tree WHERE parentID IS NULL );
IF @nulls>1 THEN SET trees=2;
ELSEIF @nulls=1 THEN
SET @root = ( SELECT childID FROM tree WHERE parentID IS NULL );
DELETE FROM tree WHERE childID=@root;
ELSE
SET @sql = CONCAT( 'SELECT Count(DISTINCT f.', parentcol, ') INTO @roots FROM ', edgeTable,
' f LEFT JOIN ', edgeTable, ' t ON f.', parentCol, '=', 't.', idCol,
' WHERE t.', idCol, ' IS NULL' );
PREPARE stmt FROM @sql; EXECUTE stmt; DROP PREPARE stmt;
IF @roots <> 1 THEN SET trees=@roots;
ELSE
SET @sql = CONCAT( 'SELECT DISTINCT f.', parentCol, ' INTO @root FROM ', edgeTable,
' f LEFT JOIN ', edgeTable, ' t ON f.', parentCol, '=', 't.',
idCol, ' WHERE t.', idCol, ' IS NULL' );
PREPARE stmt FROM @sql; EXECUTE stmt; DROP PREPARE stmt;
END IF;
END IF;
IF trees<>1 THEN
SET msg = IF( trees=0, "No tree found", "Table has multiple trees" );
SELECT msg AS 'Cannot continue';
ELSE  build nested sets tree
SET maxrightedge = 2 * (1 + (SELECT + COUNT(*) FROM tree));
INSERT INTO nestedsettree VALUES( 1, @root, 1, maxrightedge );
WHILE nextedge < maxrightedge DO
SET rows=(SELECT Count(*) FROM nestedsettree s JOIN tree t ON s.nodeID=t.parentID AND s.top=current);
IF rows > 0 THEN
BEGIN
INSERT INTO nestedsettree
SELECT current+1, MIN(t.childID), nextedge, NULL
FROM nestedsettree AS s
JOIN tree AS t ON s.nodeID = t.parentID AND s.top = current;
DELETE FROM tree
WHERE childID = (SELECT nodeID FROM nestedsettree WHERE top=(current+1));
SET nextedge = nextedge + 1, current = current + 1;
END;
ELSE
UPDATE nestedsettree SET rightedge=nextedge, top = top WHERE top=current;
SET nextedge=nextedge+1, current=current1;
END IF;
END WHILE;
 show result
IF (SELECT COUNT(*) FROM tree) > 0 THEN
SELECT 'Orphaned rows remain' AS 'Error';
END IF;
DROP TEMPORARY TABLE tree;
END IF;
END;
go
DELIMITER ;
CALL EdgeListToNestedSet( 'familytree', 'childID', 'parentID' );
SELECT
nodeID, PersonName(nodeID) AS Name,
ABS(top) AS 'Tree Level', leftedge AS 'Left', rightedge AS 'Right'
FROM nestedsettree
ORDER BY nodeID;
++++++
 nodeID  Name  Tree Level  Left  Right 
++++++
 1  Richard Shakespeare  1  1  46 
 2  Henry Shakespeare  2  2  43 
 3  John Shakespeare  2  44  45 
 4  Joan Shakespeare  3  3  4 
 5  Margaret Shakespeare  3  5  6 
 6  William Shakespeare  3  7  24 
 7  Gilbert Shakespeare  3  25  26 
 8  Joan Shakespeare  3  27  36 
 9  Anne Shakespeare  3  37  38 
 10  Richard Shakespeare  3  39  40 
 11  Edmond Shakespeare  3  41  42 
 12  Susana Shakespeare  4  8  13 
 13  Hamnet Shakespeare  4  14  15 
 14  Judith Shakespeare  4  16  23 
 15  William Hart  4  28  29 
 16  Mary Hart  4  30  31 
 17  Thomas Hart  4  32  33 
 18  Michael Hart  4  34  35 
 19  Elizabeth Hall  5  9  12 
 20  Shakespeare Quiney  5  17  18 
 21  Richard Quiney  5  19  20 
 22  Thomas Quiney  5  21  22 
 23  John Bernard  6  10  11 
++++++
Verify the function with a query that generates an edge list tree from a nested sets tree:
Listing 10a:
SELECT a.nodeID, b.nodeID AS parent
FROM nestedsettree AS a
LEFT JOIN nestedsettree AS b ON b.leftedge = (
SELECT MAX( leftedge )
FROM nestedsettree AS t
WHERE a.leftedge > t.leftedge AND a.leftedge < t.rightedge
)
ORDER BY a.nodeID;
For how to keep multiple trees in one table, see “Multiple trees in one table” on our Common Queries page.
In an edge list, the parent of a node is the row's parentID, and its children are the rows where that nodeID is parentID. What could be simpler? In comparison, nested sets queries for parents and their children are tortuous and slow. One way to fetch the child nodes of a given node is to INNER JOIN the nested sets tree table AS parent to itself AS child ON child.leftedge BETWEEN parent.leftedge AND parent.rightedge, then scope on the target row's leftedge and rightedge values. In the resulting list, child.nodeID values one level down occur once and are children, grandkids are two levels down and occur twice, and so on:
Listing 11 SELECT PersonName(child.nodeID) AS 'Descendants of William', COUNT(*) AS Generation FROM nestedsettree AS parent JOIN nestedsettree AS child ON child.leftedge BETWEEN parent.leftedge AND parent.rightedge WHERE parent.leftedge > 7 AND parent.rightedge < 24  William’s leftedge, rightedge GROUP BY child.nodeID; +++  Descendants of William  Generation  +++  Susana Shakespeare  1   Hamnet Shakespeare  1   Judith Shakespeare  1   Elizabeth Hall  2   Shakespeare Quiney  2   Richard Quiney  2   Thomas Quiney  2   John Bernard  3  +++
Therefore HAVING COUNT(t2.nodeID)=1
scopes
listed descendants to the children:
Listing 11a
SELECT PersonName(child.nodeID) AS 'Children of William'
FROM nestedsettree AS parent
JOIN nestedsettree AS child ON child.leftedge BETWEEN parent.leftedge AND parent.rightedge
WHERE parent.leftedge > 7 AND parent.rightedge < 24
GROUP BY child.nodeID
HAVING COUNT(child.nodeID)=1
++
 Children of William 
++
 Susana Shakespeare 
 Hamnet Shakespeare 
 Judith Shakespeare 
++
Retrieving a subtree or a subset of parents requires yet another join:
Listing 11b
SELECT Parent, Group_Concat(Child ORDER BY Child) AS Children
FROM (
SELECT master.nodeID AS Parent, child.nodeID AS Child
FROM nestedsettree AS master
JOIN nestedsettree AS parent
JOIN nestedsettree AS child ON child.leftedge BETWEEN parent.leftedge AND parent.rightedge
WHERE parent.leftedge > master.leftedge AND parent.rightedge < master.rightedge
GROUP BY master.nodeID, child.nodeID
HAVING COUNT(*)=1
) AS tmp
WHERE parent in(6,8,12,14)
GROUP BY Parent;
+++
 Parent  Children 
+++
 6  12,13,14 
 8  15,16,17,18 
 12  19 
 14  20,21,22 
+++
This takes hundreds of times longer than a query for the same info from an edge list! An aggregating version of Listing 19 is easier to write, but is an even worse performer:
Listing 11c
SELECT p.nodeID AS Parent, Group_Concat(c.nodeID) AS Children
FROM nestedsettree AS p
JOIN nestedsettree AS c
ON p.leftedge = (SELECT MAX(s.leftedge) FROM nestedsettree AS s
WHERE c.leftedge > s.leftedge AND c.leftedge < s.rightedge)
GROUP BY Parent;
+++
 Parent  Children 
+++
 1  2,3 
 2  5,6,7,8,9,10,11,4 
 6  12,13,14 
 8  15,16,17,18 
 12  19 
 14  20,21,22 
 19  23 
+++
Logic that is reciprocal to that of Listing 11a gets us the parent of a node:
retrieve its leftedge and rightedge values,
compute its level,
find the node which is one level up and has edge values outside the node's leftedge and rightedge values.
Listing 12
DROP PROCEDURE IF EXISTS ShowNestedSetParent;
DELIMITER go
CREATE PROCEDURE ShowNestedSetParent( node SMALLINT )
BEGIN
DECLARE level, thisleft, thisright SMALLINT DEFAULT 0;
 find node edges
SELECT leftedge, rightedge
INTO thisleft, thisright
FROM nestedsettree
WHERE nodeID = node;
 find node level
SELECT COUNT(n1.nodeid)
INTO level
FROM nestedsettree AS n1
INNER JOIN nestedsettree AS n2
ON n2.leftedge BETWEEN n1.leftedge AND n1.rightedge
WHERE n2.nodeid = node
GROUP BY n2.nodeid;
 find parent
SELECT
PersonName(n2.nodeid) AS Parent
FROM nestedsettree AS n1
INNER JOIN nestedsettree AS n2
ON n2.leftedge BETWEEN n1.leftedge AND n1.rightedge
WHERE n2.leftedge < thisleft AND n2.rightedge > thisright
GROUP BY n2.nodeid
HAVING COUNT(n1.nodeid)=level1;
END;
go
DELIMITER ;
CALL ShowNestedSetParent(6);
++
 Parent 
++
 Henry Shakespeare 
++
For some query problems, edge list and nested sets queries are equivalently simple. For example to find the tree root and leaves, compare Listing 6 with:
Listing 13
SELECT
name AS Paterfamilias,
IFNULL(born,'?') AS Born,
IFNULL(died,'?') AS Died
FROM nestedsettree AS t
INNER JOIN family AS f ON t.nodeID=f.ID
WHERE leftedge = 1;
++++
 Paterfamilias  Born  Died 
++++
 Richard Shakespeare  ?  1561 
++++
SELECT
name AS 'Childless Shakespeares',
IFNULL(born,'?') AS Born,
IFNULL(died,'?') AS Died
FROM nestedsettree AS t
INNER JOIN family AS f ON t.nodeID=f.ID
WHERE rightedge = leftedge + 1;
++++
 Childless Shakespeares  Born  Died 
++++
 Joan Shakespeare  1558  ? 
 Margaret Shakespeare  1562  1563 
 John Bernard  ?  1674 
 Hamnet Shakespeare  1585  1596 
 Shakespeare Quiney  1616  1617 
 Richard Quiney  1618  1639 
 Thomas Quiney  1620  1639 
 Gilbert Shakespeare  1566  1612 
 William Hart  1600  1639 
 Mary Hart  1603  1607 
 Thomas Hart  1605  1670 
 Michael Hart  1608  1618 
 Anne Shakespeare  1571  1579 
 Richard Shakespeare  1574  1613 
 Edmond Shakespeare  1580  1607 
 John Shakespeare  1530  1601 
++++
Finding subtrees in a nested sets model requires no twisted code, no stored procedure. To retrieve the
nestedsettree
nodes in William's subtree, just ask for nodes whose
leftedge values are greater, and whose rightedge values are smaller than William's:
Listing 14
SELECT PersonName(t.nodeID) AS Descendant
FROM nestedsettree AS s
JOIN nestedsettree AS t ON s.leftedge < t.leftedge AND s.rightedge > t.rightedge
JOIN family f ON s.nodeID = f.ID
WHERE f.name = 'William Shakespeare';
Finding a single path in the nested sets model is about as complicated as edge list path enumeration (Listings 8, 9):
Listing 15
SELECT
t2.nodeID AS Node,
PersonName(t2.nodeID) AS Person,
(SELECT COUNT(*)
FROM nestedsettree AS t4
WHERE t4.leftedge BETWEEN t1.leftedge AND t1.rightedge
AND t2.leftedge BETWEEN t4.leftedge AND t4.rightedge
) AS Path
FROM nestedsettree AS t1
INNER JOIN nestedsettree AS t2 ON t2.leftedge BETWEEN t1.leftedge AND t1.rightedge
INNER JOIN nestedsettree AS t3 ON t3.leftedge BETWEEN t2.leftedge AND t2.rightedge
WHERE t1.nodeID=(SELECT ID FROM family WHERE name='William Shakespeare')
AND t3.nodeID=(SELECT ID FROM family WHERE name='John Bernard');
++++
 Node  Person  Path 
++++
 6  William Shakespeare  1 
 12  Susana Shakespeare  2 
 19  Elizabeth Hall  3 
 23  John Bernard  4 
++++
Here the nested sets model shines. The arithmetic that was used to build the tree makes short work of summary queries. For example to retrieve a node list which preserves all parentchild relations, we need just two facts:
listing order is the order taken in the node walk that created the tree, i.e. leftedge,
a node's indentation depth is simply the JOIN (edge) count from root to node:
Listing 16
SELECT
CONCAT( SPACE(2*COUNT(parent.nodeid)2), PersonName(child.nodeid) )
AS 'The Shakespeare Family Tree'
FROM nestedsettree AS parent
INNER JOIN nestedsettree AS child
ON child.leftedge BETWEEN parent.leftedge AND parent.rightedge
GROUP BY child.nodeid
ORDER BY child.leftedge;
++
 The Shakespeare Family Tree 
++
 Richard Shakespeare 
 Henry Shakespeare 
 Joan Shakespeare 
 Margaret Shakespeare 
 William Shakespeare 
 Susana Shakespeare 
 Elizabeth Hall 
 John Bernard 
 Hamnet Shakespeare 
 Judith Shakespeare 
 Shakespeare Quiney 
 Richard Quiney 
 Thomas Quiney 
 Gilbert Shakespeare 
 Joan Shakespeare 
 William Hart 
 Mary Hart 
 Thomas Hart 
 Michael Hart 
 Anne Shakespeare 
 Richard Shakespeare 
 Edmond Shakespeare 
 John Shakespeare 
++
To retrieve only a subtree, add a query clause which restricts nodes to those whose edges are within the range of the parent node's left and right edge values, for example for William and his descendants...
WHERE parent.leftedge >= 7 AND
parent.rightedge <=24
Nested sets arithmetic also helps with insertions, updates and deletions, but the problem remains that changing just one node can require changing much of the tree.
Inserting a node in the tree requires at least two pieces of information: the nodeID to insert, and the nodeID of its parent. The model is normalised so the nodeID first must have been added to the parent (family) table. The algorithm for adding the node to the tree is:
increment leftedge by 2 in nodes where the rightedge value is greater than the parent's rightedge,
increment rightedge by 2 in nodes where the leftedge value is greater than the parent's leftedge,
insert a row with the given nodeID, leftedge = 1 + parent's leftedge, rightedge = 2 + parent's leftedge.
That's not difficult, but all rows will have to be updated!
Listing 17
DROP PROCEDURE IF EXISTS InsertNestedSetNode;
DELIMITER go
CREATE PROCEDURE InsertNestedSetNode( IN node SMALLINT, IN parent SMALLINT )
BEGIN
DECLARE parentleft, parentright SMALLINT DEFAULT 0;
SELECT leftedge, rightedge
INTO parentleft, parentright
FROM nestedsettree
WHERE nodeID = parent;
IF FOUND_ROWS() = 1 THEN
BEGIN
UPDATE nestedsettree
SET rightedge = rightedge + 2
WHERE rightedge > parentleft;
UPDATE nestedsettree
SET leftedge = leftedge + 2
WHERE leftedge > parentleft;
INSERT INTO nestedsettree
VALUES ( 0, node, parentleft + 1, parentleft + 2 );
END;
END IF;
END;
go
DELIMITER ;
"Sibline" or horizontal order is obviously significant in family trees, but may not be significant in other trees. Listing 17 adds the new node at the left edge of the sibline. To specify another position, modify the procedure to accept a third parameter for the nodeID which is to be to the left or right of the insertion point.
Updating a node in place requires nothing more than editing nodeID to point at a different parent row.
Deleting a node raises the problem of how to repair links severed by the deletion. In tree models of parts explosions, the item to be deleted is often replaced by a new item, so it can be treated like a simple node update. In organisational bossemployee charts, though, does a colleague move over, does a subordinate get promoted, does everybody in the subtree move up a level, or does something else happen? No formula can catch all the possibilities. Listing 18 illustrates how to handle two common scenarios, move everyone up, and move someone over. All possibilities except simple node replacement of require changes elsewhere in the tree.
Listing 18
DROP PROCEDURE IF EXISTS DeleteNestedSetNode;
DELIMITER go
CREATE PROCEDURE DeleteNestedSetNode( IN mode CHAR(7), IN node SMALLINT )
BEGIN
DECLARE thisleft, thisright SMALLINT DEFAULT 0;
SELECT leftedge, rightedge
INTO thisleft, thisright
FROM nestedsettree
WHERE nodeID = node;
IF mode = 'PROMOTE' THEN
BEGIN  Ian Holsman found these two bugs
DELETE FROM nestedsettree
WHERE nodeID = node;
UPDATE nestedsettree
SET leftedge = leftedge  1, rightedge = rightedge  1  rather than = thisleft
WHERE leftedge BETWEEN thisleft AND thisright;
UPDATE nestedsettree
SET rightedge = rightedge  2
WHERE rightedge > thisright;
UPDATE nestedsettree
SET leftedge = leftedge  2
WHERE leftedge > thisright;  rather than > thisleft
END;
ELSEIF mode = 'REPLACE' THEN
BEGIN
UPDATE nestedsettree
SET leftedge = thisleft  1, rightedge = thisright
WHERE leftedge = thisleft + 1;
UPDATE nestedsettree
SET rightedge = rightedge  2
WHERE rightedge > thisleft;
UPDATE nestedsettree
SET leftedge = leftedge  2
WHERE leftedge > thisleft;
DELETE FROM nestedsettree
WHERE nodeID = node;
END;
END IF;
END;
go
DELIMITER ;
Some nested sets queries are quicker than their edge list counterparts. Some are slower. None are faster than edge list queries using recursive CTEs. Given the concurrency nightmare that nested sets impose for inserts and deletions, it is reasonable to reserve the nested sets model for fairly static trees whose users are mostly interested in querying subtrees.
If you will be using the nested sets model, you may be converting back and forth with edge list models, so here is a simple query to build an edge list from a nested sets tree:
Listing 19
SELECT
p.nodeID AS parentID,
c.nodeID AS childID
FROM nestedsettree AS p
INNER JOIN nestedsettree AS c
ON p.leftedge = (SELECT MAX(s.leftedge)
FROM nestedsettree AS s
WHERE c.leftedge > s.leftedge
AND c.leftedge < s.rightedge)
ORDER BY p.nodeID;
Many graphs are not trees. Imagine a small airline which has just acquired licences for flights no longer than 6,000 km between Los Angeles (LAX), New York (JFK), Heathrow in London, Charles de Gaulle in Paris, AmsterdamSchiphol, Arlanda in Sweden, and HelsinkiVantaa. You have been asked to compute the shortest possible oneway routes that do not deviate more than 90° from the direction of the first hop—roughly, oneway routes and no circuits.
Airports are nodes, flights are edges, routes are paths. We will need three tables.
To identify an airport we need its code, location name, latitude and longitude. Latitude and longitude are usually given as degrees, minutes and seconds, north or south of the equator, east or west of Greenwich. To hide details that aren't directly relevant to nodes and edges, code latitude and longitude as simple reals where longitudes west of Greenwich and latitudes south of the equator are negative, whilst longitudes east of Greenwich and latitudes north of the equator are positive:
Listing 20
CREATE TABLE airports (
code char(3) NOT NULL,
city char(100) default NULL,
latitude float NOT NULL,
longitude float NOT NULL,
PRIMARY KEY (code)
) ;
INSERT INTO airports VALUES ('JFK', 'New York, NY', 40.75, 73.97);
INSERT INTO airports VALUES ('LAX', 'Los Angeles, CA', 34.05, 118.22);
INSERT INTO airports VALUES ('LHR', 'London, England', 51.5, 0.45);
INSERT INTO airports VALUES ('HEL', 'Helsinki, Finland', 60.17, 24.97);
INSERT INTO airports VALUES ('CDG', 'Paris, France', 48.86, 2.33);
INSERT INTO airports VALUES ('STL', 'St Louis, MO', 38.63, 90.2);
INSERT INTO airports VALUES ('ARN', 'Stockholm, Sweden', 59.33, 18.05);
The model attaches two weights to flights: distance and direction.
We need a method of calculating the Great Circle Distance—the geographical distance between any two cities  another natural job for a stored function. The distance calculation
converts to radians the degree coordinates of any two points on the earth's surface,
calculates the angle of the arc subtended by the two points, and
converts the result, also in radians, to surface (circumferential) kilometres (1 radian=6,378.388 km).
Listing 21
SET GLOBAL log_bin_trust_function_creators=TRUE;  since 5.0.16
DROP FUNCTION IF EXISTS GeoDistKM;
DELIMITER go
CREATE FUNCTION GeoDistKM( lat1 FLOAT, lon1 FLOAT, lat2 FLOAT, lon2 FLOAT ) RETURNS float
BEGIN
DECLARE pi, q1, q2, q3 FLOAT;
SET pi = PI();
SET lat1 = lat1 * pi / 180;
SET lon1 = lon1 * pi / 180;
SET lat2 = lat2 * pi / 180;
SET lon2 = lon2 * pi / 180;
SET q1 = COS(lon1lon2);
SET q2 = COS(lat1lat2);
SET q3 = COS(lat1+lat2);
SET rads = ACOS( 0.5*((1.0+q1)*q2  (1.0q1)*q3) );
RETURN 6378.388 * rads;
END;
go
DELIMITER ;
That takes care of flight distances. Flight
direction is, approximately, the arctangent (ATAN) of the difference
between flights.depart
and flights.arrive
latitudes and longitudes.
Now we can seed the airline's flights table with onehop flights up to 6,000 km long:
Listing 22
CREATE TABLE flights (
id INT PRIMARY KEY AUTO_INCREMENT,
depart CHAR(3),
arrive CHAR(3),
distance DECIMAL(10,2),
direction DECIMAL(10,2)
) ;
INSERT INTO flights
SELECT
NULL,
depart.code,
arrive.code,
ROUND(GeoDistKM(depart.latitude,depart.longitude,arrive.latitude,arrive.longitude),2),
ROUND(DEGREES(ATAN(arrive.latitudedepart.latitude,arrive.longitudedepart.longitude)),2)
FROM airports AS depart
INNER JOIN airports AS arrive ON depart.code <> arrive.code
HAVING Km <= 6000;
SELECT * FROM flights;
++++++
 id  depart  arrive  distance  direction 
++++++
 1  LAX  JFK  3941.18  8.61 
 2  LHR  JFK  5550.77  171.68 
 3  CDG  JFK  5837.46  173.93 
 4  STL  JFK  1408.11  7.44 
 5  JFK  LAX  3941.18  171.39 
 6  STL  LAX  2553.37  170.72 
 7  JFK  LHR  5550.77  8.32 
 8  HEL  LHR  1841.91  161.17 
 9  CDG  LHR  354.41  136.48 
 10  ARN  LHR  1450.12  157.06 
 11  LHR  HEL  1841.91  18.83 
 12  CDG  HEL  1912.96  26.54 
 13  ARN  HEL  398.99  6.92 
 14  JFK  CDG  5837.46  6.07 
 15  LHR  CDG  354.41  43.52 
 16  HEL  CDG  1912.96  153.46 
 17  ARN  CDG  1545.23  146.34 
 18  JFK  STL  1408.11  172.56 
 19  LAX  STL  2553.37  9.28 
 20  LHR  ARN  1450.12  22.94 
 21  HEL  ARN  398.99  173.08 
 22  CDG  ARN  1545.23  33.66 
++++++
The distances agree approximately with public information sources for flight lengths. For a pair of airports A and B not very near the poles, the error in calculating direction using ATAN(), is small. To remove that error, instead of ATAN() use a formula from spherical trigonometry (for example one of the formulas at http://www.dynagen.co.za/eugene/where/formula.html).
A route is a path along one or more of these edges, so flights:routes is a 1:many relationship. For simplicity it's efficient to denormalise representation of routes with a variation of the materialised path model to store all the hops of one route as a list of flights in one routes column. The column routes.route is the sequence of airports, from first departure to final arrival, the column routes.hops is the number of hops in that route, and the column routes.direction is the direction:
Listing 23
CREATE TABLE routes (
id INT PRIMARY KEY AUTO_INCREMENT,
depart CHAR(3),
arrive CHAR(3),
hops SMALLINT,
route CHAR(50),
distance DECIMAL(10,2),
direction DECIMAL(10,2)
) ;
Starting with an empty routes table, how do we populate it with the shortest routes between all ordered pairs of airports?
Insert all 1hop flights from the flights table.
Add in the set of shortest multihop routes for all pairs of airports which don't have rows in the flights table.
For 1hop flights we just write
Listing 24
INSERT INTO routes
SELECT
NULL,
depart,
arrive,
1,
CONCAT(depart,',',arrive),
distance,
direction
FROM flights;
NULL being the placeholder for the autoincrementing id column.
For multihop routes, we iteratively add in sets of all allowed 2hop, 3hop, ... nhop routes, replacing longer routes by shorter routes as we find them, until there is nothing more to add or replace. That also breaks down to two logical steps: add hops to build the set of next allowed routes, and update longer routes with shorter ones.
The set of next allowed routes is the set of shortest routes that can be built by adding, to existing routes, flights that leave from the last arrival airport of an existing route, arrive at an airport not yet in the given route, and stay within ± 90° of the route's initial compass direction. So every new route is a JOIN between routes and flights in which
depart = routes.depart,
arrive = flights.arrive,
flights.depart = routes.arrive,
distance = MIN(routes.distance + flights.distance),
LOCATE( flights.arrive,routes.route) = 0,
flights.direction+360 > routes.direction+270 AND flights.direction+360 < routes.direction+450
This looks like a natural logical unit of work for a View:
Listing 25
CREATE OR REPLACE VIEW nextroutes AS
SELECT
routes.depart, flights.arrive, routes.hops+1 AS hops,
CONCAT(routes.route, ',', flights.arrive) AS route,
MIN(routes.distance + flights.distance) AS distance, routes.direction
FROM routes
JOIN flights ON routes.arrive = flights.depart AND LOCATE(flights.arrive,routes.route) = 0
WHERE flights.direction BETWEEN routes.direction90 AND routes.direction+90
GROUP BY depart,arrive;
How to add these new hops to routes? In standard SQL, this variant on a query by Scott Stephens should do it...
Listing 26
INSERT INTO routes
SELECT NULL,depart,arrive,hops,route,distance,direction FROM nextroutes
WHERE (nextroutes.depart,nextroutes.arrive) NOT IN (
SELECT depart,arrive FROM routes
);
but MySQL does not yet support update table subqueries. No worries, rewriting the subquery as a join speeds it up:
Listing 27
INSERT INTO routes
SELECT
NULL,
nextroutes.depart,
nextroutes.arrive,
nextroutes.hops,
nextroutes.route,
nextroutes.distance,
nextroutes.direction
FROM nextroutes
LEFT JOIN routes ON nextroutes.depart = routes.depart
AND nextroutes.arrive = routes.arrive
WHERE routes.depart IS NULL AND routes.arrive IS NULL;
Running that code right after the initial seeding from flights gives ...
SELECT * FROM routes;
++++++++
 id  depart  arrive  hops  route  distance  direction 
++++++++
 1  LAX  JFK  1  LAX,JFK  3941.18  8.61 
 2  LHR  JFK  1  LHR,JFK  5550.77  171.68 
 3  CDG  JFK  1  CDG,JFK  5837.46  173.93 
 4  STL  JFK  1  STL,JFK  1408.11  7.44 
 5  JFK  LAX  1  JFK,LAX  3941.18  171.39 
 6  STL  LAX  1  STL,LAX  2553.37  170.72 
 7  JFK  LHR  1  JFK,LHR  5550.77  8.32 
 8  HEL  LHR  1  HEL,LHR  1841.91  161.17 
 9  CDG  LHR  1  CDG,LHR  354.41  136.48 
 10  ARN  LHR  1  ARN,LHR  1450.12  157.06 
 11  LHR  HEL  1  LHR,HEL  1841.91  18.83 
 12  CDG  HEL  1  CDG,HEL  1912.96  26.54 
 13  ARN  HEL  1  ARN,HEL  398.99  6.92 
 14  JFK  CDG  1  JFK,CDG  5837.46  6.07 
 15  LHR  CDG  1  LHR,CDG  354.41  43.52 
 16  HEL  CDG  1  HEL,CDG  1912.96  153.46 
 17  ARN  CDG  1  ARN,CDG  1545.23  146.34 
 18  JFK  STL  1  JFK,STL  1408.11  172.56 
 19  LAX  STL  1  LAX,STL  2553.37  9.28 
 20  LHR  ARN  1  LHR,ARN  1450.12  22.94 
 21  HEL  ARN  1  HEL,ARN  398.99  173.08 
 22  CDG  ARN  1  CDG,ARN  1545.23  33.66 
 23  ARN  JFK  2  ARN,LHR,JFK  7000.89  157.06 
 24  CDG  LAX  2  CDG,JFK,LAX  9778.64  173.93 
 25  CDG  STL  2  CDG,JFK,STL  7245.57  173.93 
 26  HEL  JFK  2  HEL,LHR,JFK  7392.68  161.17 
 27  JFK  ARN  2  JFK,LHR,ARN  7000.89  8.32 
 28  JFK  HEL  2  JFK,LHR,HEL  7392.68  8.32 
 29  LAX  CDG  2  LAX,JFK,CDG  9778.64  8.61 
 30  LAX  LHR  2  LAX,JFK,LHR  9491.95  8.61 
 31  LHR  LAX  2  LHR,JFK,LAX  9491.95  171.68 
 32  LHR  STL  2  LHR,JFK,STL  6958.88  171.68 
 33  STL  CDG  2  STL,JFK,CDG  7245.57  7.44 
 34  STL  LHR  2  STL,JFK,LHR  6958.88  7.44 
++++++++
... adding 12 twohop rows.
As we build routes with more hops, it is logically possible that the nextroutes view will find shorter routes for an existing routes pair of depart and arrive. Standard SQL for replacing existing routes rows with nextroutes rows that match (depart, arrive) and have shorter distance values would be ...
Listing 28
UPDATE routes SET (hops,route,distance,direction) = (
SELECT hops, route, distance, direction
FROM nextroutes
WHERE nextroutes.depart = routes.depart AND nextroutes.arrive = routes.arrive
)
WHERE (depart,arrive) IN (
SELECT depart,arrive FROM nextroutes
WHERE nextroutes.distance < routes.distance
);
... but MySQL does not support SET(col1,...) syntax, and as with Listing 7, MySQL does not yet accept subqueries referencing the table being updated, so we have to write more literal SQL:
Listing 29
UPDATE routes, nextroutes
SET
routes.hops=nextroutes.hops,
routes.route=nextroutes.route,
routes.distance=nextroutes.distance,
routes.direction=nextroutes.direction
WHERE routes.arrive=nextroutes.arrive
AND routes.depart=nextroutes.depart
AND nextroutes.distance < routes.distance;
Running this code right after the first run of Listing 27 updates no rows. To test the logic of iteration, continue running Listings 27 and 29 until no rows are being added or changed. The final result is:
SELECT * FROM ROUTES;
++++++++
 id  depart  arrive  hops  route  distance  direction 
++++++++
 1  LAX  JFK  1  LAX,JFK  3941.18  8.61 
 2  LHR  JFK  1  LHR,JFK  5550.77  171.68 
 3  CDG  JFK  1  CDG,JFK  5837.46  173.93 
 4  STL  JFK  1  STL,JFK  1408.11  7.44 
 5  JFK  LAX  1  JFK,LAX  3941.18  171.39 
 6  STL  LAX  1  STL,LAX  2553.37  170.72 
 7  JFK  LHR  1  JFK,LHR  5550.77  8.32 
 8  HEL  LHR  1  HEL,LHR  1841.91  161.17 
 9  CDG  LHR  1  CDG,LHR  354.41  136.48 
 10  ARN  LHR  1  ARN,LHR  1450.12  157.06 
 11  LHR  HEL  1  LHR,HEL  1841.91  18.83 
 12  CDG  HEL  1  CDG,HEL  1912.96  26.54 
 13  ARN  HEL  1  ARN,HEL  398.99  6.92 
 14  JFK  CDG  1  JFK,CDG  5837.46  6.07 
 15  LHR  CDG  1  LHR,CDG  354.41  43.52 
 16  HEL  CDG  1  HEL,CDG  1912.96  153.46 
 17  ARN  CDG  1  ARN,CDG  1545.23  146.34 
 18  JFK  STL  1  JFK,STL  1408.11  172.56 
 19  LAX  STL  1  LAX,STL  2553.37  9.28 
 20  LHR  ARN  1  LHR,ARN  1450.12  22.94 
 21  HEL  ARN  1  HEL,ARN  398.99  173.08 
 22  CDG  ARN  1  CDG,ARN  1545.23  33.66 
 23  ARN  JFK  2  ARN,LHR,JFK  7000.89  157.06 
 24  CDG  LAX  2  CDG,JFK,LAX  9778.64  173.93 
 25  CDG  STL  2  CDG,JFK,STL  7245.57  173.93 
 26  HEL  JFK  2  HEL,LHR,JFK  7392.68  161.17 
 27  JFK  ARN  2  JFK,LHR,ARN  7000.89  8.32 
 28  JFK  HEL  2  JFK,LHR,HEL  7392.68  8.32 
 29  LAX  CDG  2  LAX,JFK,CDG  9778.64  8.61 
 30  LAX  LHR  2  LAX,JFK,LHR  9491.95  8.61 
 31  LHR  LAX  2  LHR,JFK,LAX  9491.95  171.68 
 32  LHR  STL  2  LHR,JFK,STL  6958.88  171.68 
 33  STL  CDG  2  STL,JFK,CDG  7245.57  7.44 
 34  STL  LHR  2  STL,JFK,LHR  6958.88  7.44 
 35  ARN  LAX  3  ARN,LHR,JFK,LAX  10942.07  157.06 
 36  ARN  STL  3  ARN,LHR,JFK,STL  8409.00  157.06 
 37  HEL  LAX  3  HEL,LHR,JFK,LAX  11333.86  161.17 
 38  HEL  STL  3  HEL,LHR,JFK,STL  8800.79  161.17 
 39  LAX  ARN  3  LAX,JFK,CDG,ARN  10942.07  8.61 
 40  LAX  HEL  3  LAX,JFK,CDG,HEL  11333.86  8.61 
 41  STL  ARN  3  STL,JFK,CDG,ARN  8409.00  7.44 
 42  STL  HEL  3  STL,JFK,CDG,HEL  8800.79  7.44 
++++++++
All that's left to do is to assemble the code in a stored procedure:
Listing 30
DROP PROCEDURE IF EXISTS BuildRoutes;
DELIMITER go
CREATE PROCEDURE BuildRoutes()
BEGIN
DECLARE rows INT DEFAULT 0;
TRUNCATE routes;
 STEP 1, LISTING 24: SEED ROUTES WITH 1HOP FLIGHTS
INSERT INTO routes (depart, arrive, hops, route, distance, direction )
SELECT depart, arrive, 1, CONCAT(depart,',',arrive), distance, direction
FROM flights;
SET rows = ROW_COUNT();
WHILE (rows > 0) DO
 STEP 2, LISTINGS 25, 27: ADD NEXT SET OF ROUTES
INSERT INTO routes (depart, arrive, hops, route, distance, direction )
SELECT nextroutes.depart, nextroutes.arrive, nextroutes.hops,
nextroutes.route, nextroutes.distance, nextroutes.direction
FROM nextroutes
LEFT JOIN routes ON nextroutes.depart=routes.depart AND nextroutes.arrive=routes.arrive
WHERE routes.ID IS NULL;
SET rows = ROW_COUNT();
 STEP 3, UPDATE SHORTER ROUTES IF ANY
UPDATE routes
JOIN nextroutes USING(arrive,depart)
SET routes.hops=nextroutes.hops, routes.route=nextroutes.route,
routes.distance=nextroutes.distance, routes.direction=nextroutes.direction
WHERE nextroutes.distance < routes.distance;
END WHILE;
END;
go
DELIMITER ;
The procedure looks like a candidate for translation a CTE, but the update command and the
two joins to the table being written to (one in the nextroutes
View, one in
the insert loop) defeat the CTE engines in both MariaDB and PostgreSQL.
Route queries are straightforward. How do we check that the algorithm produced no duplicate departarrive pairs? The following query should yield zero rows ...
Listing 31
SELECT depart, arrive, COUNT(*)
FROM routes
GROUP BY depart,arrive
HAVING COUNT(*) > 1;
... and does. Reachability queries are just as simple, for example where can we fly to from Helsinki?
Listing 32
SELECT *
FROM routes
WHERE depart='HEL'
ORDER BY distance;
++++++++
 id  depart  arrive  hops  route  distance  direction 
++++++++
 21  HEL  ARN  1  HEL,ARN  398.99  173.08 
 8  HEL  LHR  1  HEL,LHR  1841.91  161.17 
 16  HEL  CDG  1  HEL,CDG  1912.96  153.46 
 26  HEL  JFK  2  HEL,LHR,JFK  7392.68  161.17 
 38  HEL  STL  3  HEL,LHR,JFK,STL  8800.79  161.17 
 37  HEL  LAX  3  HEL,LHR,JFK,LAX  11333.86  161.17 
++++++++
An extended edge list model is simple to implement, gracefully accepts extended attributes for nodes, edge and paths, does not unduly penalise updates, and responds to queries with reasonable speed.
A bill of materials for a house would include the cement block, lumber, shingles, doors, wallboard, windows, plumbing, electrical system, heating system, and so on. Each subassembly also has a bill of materials; the heating system has a furnace, ducts, and so on. A bill of materials implosion links component pieces to a major assembly. A bill of materials explosion breaks apart assemblies and subassemblies into their component parts.
Which graph model best handles a parts explosion? Combining edge list and "nested sets" algorithms seems a natural solution.
Imagine a new company that plans to make variously sized bookcases, either packaged as doityourself kits of, or assembled from sides, shelves, shelf brackets, backboards, feet and screws. Shelves and sides are cut from planks. Backboards are trimmed from laminated sheeting. Feet are machinecarved from readycut blocks. Screws and shelf brackets are purchased in bulk. Here are the elements of one bookcase ...
1 backboard, 2 x 1 m
1 laminate
8 screws
2 sides 2m x 30 cm
1 plank length 4m
12 screws
8 shelves 1 m x 30 cm (incl. top and bottom)
2 planks
24 shelf brackets
4 feet 4cm x 4cm
4 cubes
16 screws
... which may be packaged in a box for sale at one price, or assembled as a finished product at a different price. At any time we need to be able to answer questions like
Do we have enough parts to make the bookcases on order?
What assemblies or packages would be most profitable to make given the current inventory?
There is no reason to break the normalising rule that item detail belongs in a nodes table, and graph logic belongs in an edges table. Edges also require a quantity attribute, for example a shelf includes four shelf brackets. Nodes and edges may also have costs and prices:
item purchase cost,
item assembly cost,
assembly cost,
assembly selling price.
In many parts problems like this one, items occur in multiple assemblies and subassemblies. The graph is not a tree. Also, it is often desirable to model multiple graphs without the table glut that would arise from giving each graph its own edges table. A simple way to solve this problem is to represent multiple graphs (assemblies) in the edges table by giving every row not only childID and parentID pointers, but a pointer which identifies the root itemID of the graph to which the row belongs.
So the data model is just two tables, for items (nodes and for product graphs or assemblies (edges). Assume that the company begins with a plan to sell the 2m x 1m bookcase in two forms, assembled and kit, and that the purchasing department has bought quantities of raw materials (laminate, planks, shelf supports, screws, wood cubes, boxes). Here are the nodes (items) and edges (assemblies):
Listing 33
CREATE TABLE items (
itemID INT PRIMARY KEY AUTO_INCREMENT,
name CHAR(20) NOT NULL,
onhand INT NOT NULL DEFAULT 0,
reserved INT NOT NULL DEFAULT 0,
purchasecost DECIMAL(10,2) NOT NULL DEFAULT 0,
assemblycost DECIMAL(10,2) NOT NULL DEFAULT 0,
price DECIMAL(10,2) NOT NULL DEFAULT 0
);
CREATE TABLE assemblies (
assemblyID INT NOT NULL,
assemblyroot INT NOT NULL,
childID INT NOT NULL,
parentID INT NOT NULL,
quantity DECIMAL(10,2) NOT NULL,
assemblycost DECIMAL(10,2) NOT NULL,
PRIMARY KEY(assemblyID,childID,parentID)
);
INSERT INTO items VALUES  inventory
(1,'laminate',40,0,4,0,8),
(2,'screw',1000,0,0.1,0,.2),
(3,'plank',200,0,10,0,20),
(4,'shelf bracket',400,0,0.20,0,.4),
(5,'wood cube',100,0,0.5,0,1),
(6,'box',40,0,1,0,2),
(7,'backboard',0,0,0,3,0),
(8,'side',0,0,0,8,0),
(9,'shelf',0,0,0,4,0),
(10,'foot',0,0,0,1,0),
(11,'bookcase2x30',0,0,0,10,0),
(12,'bookcase2x30 kit',0,0,0,2,0);
INSERT INTO assemblies VALUES
(1,11,1,7,1,0),  laminate to backboard
(2,11,2,7,8,0),  screws to backboard
(3,11,3,8,.5,0),  planks to side
(4,11,2,8,6,0),  screws to side
(5,11,3,9,0.25,0),  planks to shelf
(6,11,4,9,4,0),  shelf brackets to shelf
(7,11,5,10,1,0),  wood cubes to foot
(8,11,2,10,1,0),  screws to foot
(9,11,7,11,1,0),  backboard to bookcase
(10,11,8,11,2,0),  sides to bookcase
(11,11,9,11,8,0),  shelves to bookcase
(12,11,10,11,4,0),  feet to bookcase
(13,12,1,7,1,0),  laminate to backboard
(14,12,2,7,8,0),  screws to backboard
(15,12,3,8,0.5,0),  planks to side
(16,12,2,8,6,0),  screws to sides
(17,12,3,9,0.25,0),  planks to shelf
(18,12,4,9,4,0),  shelf brackets to shelves
(19,12,5,10,1,0),  wood cubes to foot
(20,12,2,10,1,0),  screws to foot
(21,12,7,12,1,0),  backboard to bookcase kit
(22,12,8,12,2,0),  sides to bookcase kit
(23,12,9,12,8,0),  shelves to bookcase kit
(24,12,10,12,4,0),  feet to bookcase kit
(25,12,6,12,1,0);  container box to bookcase kit
Now, we want a parts list, a bill of materials, which will list show parentchild relationships and quantities, and sum the costs. Could we adapt the depthfirst "nested sets" treewalk algorithm (Listing 10) to this problem even though our graph is not a tree and our sets are not properly nested? Yes: touch up the treewalk to handle multiple parent nodes for any child node, and add code to percolate costs and quantities up the graph. Navigation remains simple using leftedge and rightedge values. This is just the sort of problem the Celko algorithm is good for: reporting!
Listing 34
CREATE PROCEDURE ShowBOM( IN root INT )
BEGIN
DECLARE thischild, thisparent, rows, maxrightedge INT DEFAULT 0;
DECLARE thislevel, nextedgenum INT DEFAULT 1;
DECLARE thisqty, thiscost DECIMAL(10,2);
 Create and seed intermediate table:
DROP TABLE IF EXISTS edges;
CREATE TABLE edges (
childID smallint NOT NULL,
parentID smallint NOT NULL,
PRIMARY KEY (childID, parentID)
) ENGINE=HEAP;
INSERT INTO edges
SELECT childID,parentID
FROM assemblies
WHERE assemblyRoot = root;
SET maxrightedge = 2 * (1 + (SELECT COUNT(*) FROM edges));
 Create and seed result table:
DROP TABLE IF EXISTS bom;
CREATE TABLE bom (
level SMALLINT,
nodeID SMALLINT,
parentID SMALLINT,
qty DECIMAL(10,2),
cost DECIMAL(10,2),
leftedge SMALLINT,
rightedge SMALLINT
) ENGINE=HEAP;
INSERT INTO bom VALUES( thislevel, root, 0, 0, 0, nextedgenum, maxrightedge );
SET nextedgenum = nextedgenum + 1;
WHILE nextedgenum < maxrightedge DO
 How many children of this node remain in the edges table?
SET rows = (
SELECT COUNT(*)
FROM bom AS p
JOIN edges AS c ON p.nodeID=c.parentID AND p.level=thislevel
);
IF rows > 0 THEN
 Child edge exists. Compute qty & cost, insert in bom, delete from edges.
BEGIN
 Alas MySQL nulls MIN(t.childid) when we combine the next two queries
SET thischild = (
SELECT MIN(c.childID)
FROM bom AS p
INNER JOIN edges AS c ON p.nodeID=c.parentID AND p.level=thislevel
);
SET thisparent = (
SELECT DISTINCT c.parentID
FROM bom AS p
INNER JOIN edges AS c ON p.nodeID=c.parentID AND p.level=thislevel
);
SET thisqty = (
SELECT quantity FROM assemblies
WHERE assemblyroot = root
AND childID = thischild
AND parentID = thisparent
);
SET thiscost = (
SELECT thisqty * ( a.assemblycost + i.purchasecost + i.assemblycost )
FROM assemblies AS a
JOIN items AS i ON a.childID = i.itemID
WHERE assemblyroot = root
AND a.parentID = thisparent
AND a.childID = thischild
);
INSERT INTO bom
VALUES(thislevel+1, thischild, thisparent, thisqty, thiscost, nextedgenum, NULL);
DELETE FROM edges WHERE childID=thischild AND parentID=thisparent;
SET thislevel = thislevel + 1, nextedgenum = nextedgenum + 1;
END;
ELSE
BEGIN
 Set rightedge, remove item from edges
UPDATE bom
SET rightedge=nextedgenum, level = level
WHERE level = thislevel;
SET thislevel = thislevel – 1, nextedgenum = nextedgenum + 1;
END;
END IF;
END WHILE;
SET rows := ( SELECT COUNT(*) FROM edges );
IF rows > 0 THEN
SELECT 'Orphaned rows remain';
ELSE
BEGIN
SET thiscost = (SELECT SUM(cost*qty) FROM bom);
UPDATE bom SET qty=1, cost=thiscost WHERE nodeID = root;
SELECT
CONCAT(Space(Abs(level)*2), ItemName(nodeid,root)) AS Item,
ROUND(qty,1) AS Qty,
ROUND(cost,2) AS Cost
FROM bom
ORDER BY leftedge;
END;
END IF;
END;
go
DELIMITER ;
 Function used by ShowBOM() to retrieve bom item names:
DROP FUNCTION IF EXISTS ItemName;
SET GLOBAL log_bin_trust_function_creators=TRUE;
DELIMITER go
CREATE FUNCTION ItemName( id INT, root INT ) RETURNS CHAR(20)
BEGIN
DECLARE s CHAR(20) DEFAULT '';
SELECT name INTO s FROM items WHERE itemid=id;
RETURN IF( id = root, UCASE(s), s );
END;
go
DELIMITER ;
CALL SHOWBOM(11);
++++
 Item  Qty  Cost 
++++
 BOOKCASE2X30  1.0  327.93 
 backboard  1.0  3.00 
 laminate  1.0  4.00 
 screw  8.0  0.80 
 side  2.0  16.00 
 screw  6.0  0.60 
 plank  0.5  5.00 
 shelf  8.0  32.00 
 plank  0.3  2.50 
 shelf bracket  4.0  0.80 
 foot  4.0  4.00 
 screw  1.0  0.10 
 wood cube  1.0  0.50 
++++
With ShowBOM() in hand, it's easy to compare costs of assemblies and subassemblies. By adding price columns, we can do the same for prices and profit margins. And now that MySQL has reenabled prepared statements in stored procedures, it will be relatively easy to write a more general version of ShowBOM(). We leave that to you.
But ShowBOM() is not the small, efficient
bit of nested sets reporting code we were hoping for. There is a
simpler solution: hide graph cycles from the edges table by making
them references to rows in a nodes table, so we can treat the edges
table like a tree; then apply a breadthfirst edgelist
subtree algorithm to generate the Bill of Materials. Again assume
a cabinetmaking company making bookcases (with a different costing
model). For clarity, skip inventory tracking for now. An items table
ww_nodes
tracks purchased and assembled bookcase elements with their
individual costs, and an assemblies/edges ww_edges table tracks sets
of edges that combine to make products.
Listing 35: DDL for a simpler parts explosion
DROP TABLE IF EXISTS ww_nodes;
CREATE TABLE ww_nodes (
nodeID int,
description CHAR(50),
cost decimal(10,2)
);
INSERT INTO ww_nodes VALUES (1,'finished bookcase',10);
INSERT INTO ww_nodes VALUES (2,'backboard2x1',1);
INSERT INTO ww_nodes VALUES (3,'laminate2x1',8);
INSERT INTO ww_nodes VALUES (4,'screw',.10);
INSERT INTO ww_nodes VALUES (5,'side',4);
INSERT INTO ww_nodes VALUES (6,'plank',20);
INSERT INTO ww_nodes VALUES (7,'shelf',4);
INSERT INTO ww_nodes VALUES (8,'shelf bracket',.5);
INSERT INTO ww_nodes VALUES (9,'feet',1);
INSERT INTO ww_nodes VALUES (10,'cube4cmx4cm',1);
INSERT INTO ww_nodes VALUES (11,'bookcase kit',2);
INSERT INTO ww_nodes VALUES (12,'carton',1);
DROP TABLE IF EXISTS ww_edges;
CREATE TABLE ww_edges (
rootID INT,
nodeID int,
parentnodeID int,
qty decimal(10,2)
);
INSERT INTO ww_edges VALUES (1,1,null,1);
INSERT INTO ww_edges VALUES (1,2,1,1);
INSERT INTO ww_edges VALUES (1,3,2,1);
INSERT INTO ww_edges VALUES (1,4,2,8);
INSERT INTO ww_edges VALUES (1,5,1,2);
INSERT INTO ww_edges VALUES (1,6,5,1);
INSERT INTO ww_edges VALUES (1,4,5,12);
INSERT INTO ww_edges VALUES (1,7,1,8);
INSERT INTO ww_edges VALUES (1,6,7,.5);
INSERT INTO ww_edges VALUES (1,8,7,4);
INSERT INTO ww_edges VALUES (1,9,1,4);
INSERT INTO ww_edges VALUES (1,10,9,1);
INSERT INTO ww_edges VALUES (1,4,9,1);
INSERT INTO ww_edges VALUES (11,11,null,1);
INSERT INTO ww_edges VALUES (11,2,11,1);
INSERT INTO ww_edges VALUES (11,3,2,1);
INSERT INTO ww_edges VALUES (11,4,2,8);
INSERT INTO ww_edges VALUES (11,5,11,2);
INSERT INTO ww_edges VALUES (11,6,5,1);
INSERT INTO ww_edges VALUES (11,4,5,12);
INSERT INTO ww_edges VALUES (11,7,11,8);
INSERT INTO ww_edges VALUES (11,6,7,.5);
INSERT INTO ww_edges VALUES (11,8,7,4);
INSERT INTO ww_edges VALUES (11,9,11,4);
INSERT INTO ww_edges VALUES (11,10,9,1);
INSERT INTO ww_edges VALUES (11,4,9,11);
INSERT INTO ww_edges VALUES (11,12,11,1);
Here is an adaptation of the breadthfirst edge list algorithm to retrieve a Bill of Materials for a product identified by a rootID:
1. Initialise a leveltracking variable to zero.
2. Seed a temp reporting table with the rootID of the desired product.
3. While rows are retrieved, increment level
and
add rows to the temp table whose parentnodeIDs are nodes at the current level.
4. Print the BOM ordered by path with indentation proportional to tree level.
Listing 36: A simpler parts explosion DROP PROCEDURE IF EXISTS ww_bom; DELIMITER go CREATE PROCEDURE ww_bom( root INT ) BEGIN DECLARE lev INT DEFAULT 0; DECLARE totalcost DECIMAL( 10,2); DROP TABLE IF EXISTS temp; CREATE TABLE temp  initialise temp table with root node SELECT e.nodeID AS nodeID, n.description AS Item, e.parentnodeID, e.qty, n.cost AS nodecost, e.qty * n.cost AS cost, 0 as level,  tree level CONCAT(e.nodeID,'') AS path  path to this node as a string FROM ww_nodes n JOIN ww_edges e USING(nodeID)  root node WHERE e.nodeID = root AND e.parentnodeID IS NULL; WHILE FOUND_ROWS() > 0 DO BEGIN SET lev = lev+1;  increment level INSERT INTO temp  add children of this level SELECT e.nodeID, n.description AS Item, e.parentnodeID, e.qty, n.cost AS nodecost, e.qty * n.cost AS cost, lev, CONCAT(t.path,',',e.nodeID) FROM ww_nodes n JOIN ww_edges e USING(nodeID) JOIN temp t ON e.parentnodeID = t.nodeID WHERE e.rootID = root AND t.level = lev1; END; END WHILE; WHILE lev > 0 DO  percolate costs up the graph BEGIN SET lev = lev  1; DROP TABLE IF EXISTS tempcost; CREATE TABLE tempcost  compute child cost SELECT p.nodeID, SUM(c.nodecost*c.qty) AS childcost FROM temp p JOIN temp c ON p.nodeid=c.parentnodeid WHERE c.level=lev GROUP by p.nodeid; UPDATE temp JOIN tempcost USING(nodeID)  update parent item cost SET nodecost = nodecost + tempcost.childcost; UPDATE temp SET cost = qty * nodecost  update parent cost WHERE level=lev1; END; END WHILE; SELECT  list BoM CONCAT(SPACE(level*2),Item) AS Item, ROUND(nodecost,2) AS 'Unit Cost', ROUND(Qty,0) AS Qty,ROUND(cost,2) AS Cost FROM temp ORDER by path; END; go DELIMITER ; CALL ww_bom( 1 ); +++++  Item  Unit Cost  Qty  Cost  +++++  finished bookcase  206.60  1.0  206.60   backboard2x1  9.80  1.0  9.80   laminate2x1  8.00  1.0  8.00   screw  0.10  8.0  0.80   side  25.20  2.0  50.40   screw  0.10  12.0  1.20   plank  20.00  1.0  20.00   shelf  16.00  8.0  128.00   plank  20.00  0.5  10.00   shelf bracket  0.50  4.0  2.00   foot  2.10  4.0  8.40   cube4cmx4cm  1.00  1.0  1.00   screw  0.10  1.0  0.10  +++++
Stored procedures, stored functions and Views make it possible to implement edge list graph models, nested sets graph models, and breadthfirst and depthfirst graph search algorithms in MySQL 5.x. Common Table Expressions in MariaDB since version 10.2.2, and in MySQL since version 8.0.1, greatly improve edge list query performance. .
Bichot G, “MySQL 8.0.1: [Recursive] Common Table Expressions in MySQL (CTEs), Part Four – depthfirst or breadthfirst traversal, transitive closure, cycle avoidance”, http://mysqlserverteam.com/mysql801recursivecommontableexpressionsinmysqlctespartfourdepthfirstorbreadthfirsttraversaltransitiveclosurecycleavoidance/
Celko J, "Trees and Hierarchies in SQL For Smarties", Morgan Kaufman, San Francisco, 2004.
Codersource.net, "Branch and Bound Algorithm in C#", http://www.codersource.net/csharp_branch_and_bound_algorithm_implementation.aspx.
Math Forum, "Euler's Solution: The Degree of a Vertex", http://mathforum.org/isaac/problems/bridges2.html
Muhammad RB, "Trees", http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/trees.htm.
Mullins C, "The Future of SQL", http://www.craigsmullins.com/idug_sql.htm.
Murphy K, "A Brief Introduction to Graphical Models and Bayesian Networks", http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html
Rodrigue JP, "Graph Theory: Definition and Properties", http://people.hofstra.edu/geotrans/eng/ch2en/meth2en/ch2m1en.html.
Santry P, "Recursive SQL User Defined Functions", http://www.wwwcoder.com/main/parentid/191/site/1857/68/default.aspx.
Shasha D, Wang JTL, and Giugno R, "Algorithmics and applications of tree and graph searching", In Symposium on Principles of Database Systems, 2002, p 3952.
Stephens S, "Solving directed graph problems with SQL, Part I", http://builder.com.com/51006388_145245017.html.
Stephens, S, "Solving directed graph problems with SQL, Part II", http://builder.com.com/51006388_145253701.html.
Steinbach T, "Migrating Recursive SQL from Oracle to DB2 UDB", http://www106.ibm.com/developerworks/db2/library/techarticle/0307steinbach/0307steinbach.html.
Tropashko V, "Nested Intervals Tree Encoding in SQL, http://www.sigmod.org/sigmod/record/issues/0506/p47articletropashko.pdf
Van Tulder G, "Storing Hierarchical Data in a Database", http://www.sitepoint.com/print/hierarchicaldatadatabase.
Venagalla S, "Expanding Recursive Opportunities with SQL UDFs in DB2 v7.2", http://www106.ibm.com/developerworks/db2/library/techarticle/0203venigalla/0203venigalla.html.
Wikipedia, "Graph Theory", http://en.wikipedia.org/wiki/Graph_theory.
Wikipedia, “Tree traversal”, Wikipedia,“Tree traversal”, http://en.wikipedia.org/wiki/Tree_traversal.
Willets K, "SQL Graph Algorithms", http://willets.org/sqlgraphs.html.

Last updated 13 Aug 2017 