1. Discrete Mathematics
Chapter 5
Trees

2. 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.

3. 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

4. Rooted tree A rooted tree is a tree where one of its vertices is
designated the root

5. 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

6. 7.2 Terminology

7. Root Leaf Siblings Parent of node n Child of node n
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

8. 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

9. 7.5 Binary trees
A binary tree is a tree where each vertex has zero, one or two children

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

11. 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

12. Example:
Number of nodes=7
Number of comparison=4
Element exists

13. Example:
Number of nodes=7
Number of comparison=3
Element exists

14. Example: Key=15 Number of nodes=7 Number of comparison=4
Element isn’t exists

15. Building trees:

16. Another example:

17. Operations on trees:
Insert or add on tree.

18. 2) Deletion.

19. 7.6 Tree Traversals
1: Pre-order traversal
NLR

21. 7.6 Tree Traversals
1: in-order traversal
LNR

23. More on tree traversals
3: Post-order traversal
LRN