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A tree is a special case of a grapha connected, acyclic graph. A rooted tree is accessed beginning at the root node. Each node is either a leaf or an internal node. An internal node has one or more child nodes and is called the parent of its child nodes. All children of the same node are siblings. Contrary to the appearance in a physical tree, the root is usually depicted at the top of the structure and the leaves are depicted at the bottom. (See Figure 9-1.)
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Figure 9-1. A tree
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A forest is a collection of one or more treesfor example, forum discussions can be represented as a forest where each thread is a tree.
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Some scenarios can be described as a hierarchy and modeled as an directed acyclic graphfor example, inheritance among types/classes in object-oriented programming and reports-to relationships in an employee organizational chart. In the former, the edges of the graph locate the inheritance. Classes can inherit methods and properties from other classes (and possibly from multiple classes). In the latter, the edges represent the reports-to relationship between employees. Notice the acyclic, directed nature of these scenarios. The management chain of responsibility in a company cannot go around in circles, for example.
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Throughout the chapter, I will use three scenarios: Employee Organizational Chart (tree, hierarchy), Bill Of Materials or BOM (DAG), and Road System (undirected cyclic graph). Note what distinguishes a (directed) tree from a DAG. All trees are DAGs, but not all DAGs are trees. In a tree, an item can have at most one parent; in some management hierarchies, an employee can have more than one manager.
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Employee Organizational Chart
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The employee organizational chart that I will use is depicted graphically in Figure 9-2.
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Figure 9-2. Employee Organizational Chart
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To create the Employees table and populate it with sample data, run the code in Listing 9-1. The contents of the Employees table are shown in Table 9-1.
Listing 9-1. Data definition language and sample data for the Employees table
SET NOCOUNT ON; USE tempdb; GO IF OBJECT_ID('dbo.Employees') IS NOT NULL DROP TABLE dbo.Employees; GO CREATE TABLE dbo.Employees ( empid INT NOT NULL PRIMARY KEY, mgrid INT NULL REFERENCES dbo.Employees, empname VARCHAR(25) NOT NULL, salary MONEY NOT NULL, CHECK (empid <> mgrid) ); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(1, NULL, 'David', $10000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(2, 1, 'Eitan', $7000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(3, 1, 'Ina', $7500.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(4, 2, 'Seraph', $5000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(5, 2, 'Jiru', $5500.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(6, 2, 'Steve', $4500.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(7, 3, 'Aaron', $5000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(8, 5, 'Lilach', $3500.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(9, 7, 'Rita', $3000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(10, 5, 'Sean', $3000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(11, 7, 'Gabriel', $3000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(12, 9, 'Emilia' , $2000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(13, 9, 'Michael', $2000.00); INSERT INTO dbo.Employees(empid, mgrid, empname, salary) VALUES(14, 9, 'Didi', $1500.00); CREATE UNIQUE INDEX idx_unc_mgrid_empid ON dbo.Employees(mgrid, empid);
Table 9-1. Contents of Employees Table
empid mgrid empname salary 1 NULL David 10000.0000
empid mgrid empname salary 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1 2 2 2 3 5 7 5 7 9 9 9 Eitan Ina Seraph Jiru Steve Aaron Lilach Rita Sean Gabriel Emilia Michael Didi 7000.0000 7500.0000 5000.0000 5500.0000 4500.0000 5000.0000 3500.0000 3000.0000 3000.0000 3000.0000 2000.0000 2000.0000 1500.0000
The Employees table represents a management hierarchy as an adjacency list, where the manager and employee represent the parent and child nodes, respectively.
Bill of Materials (BOM)
I will use a BOM of coffee shop products, which is depicted graphically in Figure 9-3.
Figure 9-3. Bill of Materials (BOM)
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To create the Parts and BOM tables and populate them with sample data, run the code in Listing 9-2. The contents of the Parts and BOM tables are shown in Tables 9-2 and 9-3.
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