Discrete Mathematics

Chapter 5 Trees

7.1 Introduction Tree

A tree is a connected undirected graph that contains no circuits.

Recall: A circuit is a path of length >=1 that begins and ends a the same vertex.

Free tree

A (free) tree T is A simple graph

such that for every pair of vertices v and w

there is a unique path from v to w

Rooted tree

A rooted tree is a tree

where one of its vertices is

designated the root

Level of a vertex and tree height

Let T be a rooted tree: The level l(v) of a vertex v is

the length of the simple path from v to the root of the tree

The height h of a rooted tree T is the maximum of all level numbers of its vertices:

h = max { l(v) } v V(T)

Example: the tree on the right has height 3

7.2 Terminology

Root The only node in

the tree with no parent

Leaf A node with no

children Siblings

Nodes with a common parent

Parent of node n The node directly

above node n in the tree

Child of node n A node directly

below node n in the tree

Internal and terminal vertices

An internal vertex is a vertex that has at least one child

A terminal vertex is a vertex that has no children

The tree in the example has 4 internal vertices and 4 terminal vertices

7.5 Binary trees A binary tree is a tree

where each vertex has zero, one or two children

Full binary tree

A full binary tree is a binary tree in which each vertex has two or no children.

Binary search trees Data are associated to

each vertex Order data alphabetically,

so that for each vertex v, data to the left of v are less than data in v

and data to the right of v are greater than data in v

Example:

Number of nodes=7Number of comparison=4Element exists

Example:

Number of nodes=7Number of comparison=3Element exists

Example:

Key=15Number of nodes=7Number of comparison=4Element isn’t exists

Building trees:

Another example:

Operations on trees:1)Insert or add on tree.

2) Deletion.

7.6 Tree Traversals

1: Pre-order traversal

NLR

7.6 Tree Traversals

1: in-order traversal

LNR

More on tree traversals 3: Post-order traversal

LRN