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### Lecture 8Tree

Tree

Tree:

Tree is a connected graph with no cycle

Theorem:

Let G be graph with more than one vertex. Then the following are equivalence:

1) G is a tree.

2) G is cycle-free with (n-1) edges.

3) G is connected and has (n-1) edges. (i.e: if any edge is deleted then the resulting graph is not connected)

Rooted tree:

A rooted tree R consists of a tree graph together with vertex r called the root of the tree. ( is a directed tree where all arrows are directed a way from a single vertex)

Tree

Height or depth: The number of levels of a tree

Leaves: The vertices of the tree that have no child (vertices with degree one)

Degree of tree: The largest number of children in the vertices of the tree

Binary tree : every vertex has at most 2 children

Order Rooted Tree (ORT): Whenever draw the digraph of a tree, we assume some ordering at each level, by arranging children from left to right.

Or ; is a binary tree such that the set of children of each parent is linearly ordered from left to right.

Any algebraic expression involving binary operations +, -, , ÷ can be represented by an order rooted tree (ORT)

the binary rooted tree for a+b is :

The variable in the expression a & b appear as leaves and the operations appear as the other vertices.

Tree , ÷ can be represented by an order rooted tree (ORT)

Polish notation (Binary Tree Traversal Algorithm):

The polish notation form of an algebraic expression represents the expression unambiguously with out the need for parentheses

1) In order LOR (infix) : a + b

2) Pre order OLR(prefix) : + a b

3) Post order LRO (postfix) : a b +

Tree , ÷ can be represented by an order rooted tree (ORT)

example 1: infix polish notation is : a + b

prefix polish notation : + a b

example 2: infix polish notation is : a + 2 * b

prefix polish notation : + a * 2 b

Example5: according to the following digraph , ÷ can be represented by an order rooted tree (ORT)

Post order : 1.5.3.15.30.20.9.8

Pre order : 8.3.1.5.9.20.15.30

In order : 1.3.5.8.9.15.20.30

8

9

3

20

5

1

30

15

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