Chap 8 Trees

• Def 1: A tree is a connected,undirected, graph with no simple circuits .

Ex1.

• Theorem1: An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.

Chap 8 Trees

• Root , rooted tree, parent , child , siblings , ancestors , descendants,leaf ,subtree,internal vertices:have children

• Ex2

• Def 2: m-ary tree : every internal vertex has no more than m children;full m-ary tree ; binary tree.

Chap 8 Trees

• Trees as models

• Properties of trees

• Theorem2: A tree with n vertices has n-1 edges.

• Theorem3: A full m-ary tree with i internal vertices contains n=mi+1 vertices.

Chap 8 Trees

• Level of a vertex :length of the unique path from the root to the vertex

• height of a rooted tree:length of the longest path from the root

• Ex10 a rooted m-ary tree of height h is balance if all leaves are at levels h or h-1

Chap 8 Trees

• Tree traversal ordered rooted tree ; left/right child/subtree• Def1: preorder traversal of an ordered rooted tree Fig 2; Ex2 Def 2: inorder traversal Fig 5, Ex3 Def 3: postorder traversal Fig 7, Ex4

Chap 8 Trees

• Fig 9 :

preorder : list each vertex the first time this

curve passes it .

inorder : list a leaf the first time the curve

passes it ; list each internal vertex the

second time the curve passes it postorder :list a vertex the last time it is passed

on the way back up to its parent.

Chap 8 Trees

• Represent complicated expressions using ordered rooted trees

• Ex5 inorder traversal produces the original expression

with the elements and operations in the same order as they originally occurred .

infix form ( Fig 11) : need to include parentheses

whenever an operation is encountered in the inorder traversal

Chap 8 Trees

• Prefix form : no parenthesis are needed (Polish notation)

Ex6

We can evaluate an expression in prefix

form by working from right to left

Ex7

Postfix form (reverse Polish notation)

: no parenthesis are needed

Chap 8 Trees• Ex8

evaluate an expression from left to right

Ex9

Ex10

Because prefix and postfix expressions are unambiguous and can easily be evaluated , they are extensively used in computers science.

Chap 8 Trees• Spanning Tree Def1 : let G be a simple graph .A spanning tree of G is a

subgraph of G that is a tree containing every vertex of G (n vertices with n-1 edges) Ex1. Algorithm for constructing spanning tree depth-first search /backtracking Ex3. breadth-first search Ex4.

Chap 8 Trees

Minimum Spanning Tree • Def 1 : A minimum spanning tree in a connected

weighted graph is a spanning tree that has the smallest possible sum of weights of its edges

Prim’s algorithmEx2. Kruskal’s algorithmEx3.