1. Binary Trees

2. Introduction
A tree structure means that the data are organized so that items of information are related by branches
Examples:

3. Parts of a binary tree A binary tree is composed of zero or more nodes
Each node contains:
A value (some sort of data item)
A reference or pointer to a left child (may be null), and
A reference or pointer to a right child (may be null)
A binary tree may be empty (contain no nodes)
If not empty, a binary tree has a root node
Every node in the binary tree is reachable from the root node by a unique path
A node with neither a left child nor a right child is called a leaf
In some binary trees, only the leaves contain a value

4. Terminology
Some node: the item of information plus the branches to each node.
degree: the number of sub trees of a node
degree of a tree: the maximum of the degree of the nodes in the tree.
terminal nodes (or leaf): nodes that have degree zero
nonterminal nodes: nodes that don’t belong to terminal nodes.
children: the roots of the sub trees of a node X are the children of X
parent: X is the parent of its children.

5. Terminology
siblings: children of the same parent are said to be siblings.
Ancestors of a node: all the nodes along the path from the root to that node.
The level of a node: defined by letting the root be at level one. If a node is at level l, then it children are at level l+1.
Height (or depth): the maximum level of any node in the tree

6. Picture of a binary treed e g h i l f j k

7. Size and depth The size of a binary tree is the number of nodes in it
This tree has size 12
The depth of a node is its distance from the root
a is at depth zero
e is at depth 2
The depth of a binary tree is the depth of its deepest node
This tree has depth 4 a b c d e f g h i j k l

8. Balance a b c d e f g h i j A balanced binary tree a b c d e f g h i j
An unbalanced binary tree
A binary tree is balanced if every level above the lowest is “full” (contains 2n nodes)
In most applications, a reasonably balanced binary tree is desirable

9. Example Property: (# edges) = (#nodes) - 1 A is the root node
B is the parent of D and E
C is the sibling of B
D and E are the children of B
D, E, F, G, I are external nodes, or leaves
A, B, C, H are internal nodes
The level of E is 3
The height (depth) of the tree is 4
The degree of node B is 2
The degree of the tree is 3
The ancestors of node I is A, C, H
The descendants of node C is F, G, H, I
Property: (# edges) = (#nodes) - 1 A B C H I D E F G
Level 1 2 3 4

10. Example of a Tree No. of nodes = 9 Height = 4 Highest Level = 3
5/16/2019
Example of a Tree
No. of nodes = 9
Height = 4
Highest Level = 3
Root Node = 8
Leaves = 1,4,7,13
Interior Nodes = 3,10,6,14
Ancestors Of 6 = 3,8
Descendents Of 10 = 14,13
Sibling Of 1 = 6

11. Full Binary Tree and Complete Binary Tree

12. Traversal
A traversal is a process that visits all the nodes in the tree. Since a tree is a
nonlinear data structure, there is no unique traversal. We will consider
several traversal algorithms with we group in the following two kinds
Depth-first traversal.
Breadth-first traversal .
There are three different types of depth-first traversals :
PreOrder traversal - visit the parent first and then left and right children;
InOrder traversal - visit the left child, then the parent and the right child;
PostOrder traversal - visit left child, then the right child and then the parent;
There is only one kind of breadth-first traversal--the level order traversal. This traversal visits nodes by levels from top to bottom and from left to right.

13. As an example consider the following tree and its four traversals: PreOrder - 8, 5, 9, 7, 1, 12, 2, 4, 11, 3 InOrder - 9, 5, 1, 7, 2, 12, 8, 4, 3, 11 PostOrder - 9, 1, 2, 12, 7, 5, 3, 11, 4, 8

14. Binary Tree Traversals
Inorder traversal (LVR) (recursive version)
output: A / B * C * D + E
ptr L V R

15. Binary Tree Traversals
Preorder traversal (VLR) (recursive version)
output: + * * / A B C D E V L R

16. Binary Tree Traversals
Postorder traversal (LRV) (recursive version)
output: A B / C * D * E + L R V