1. Tree

2. Tree A tree is a collection of nodes and edges
A tree has only one root
Trees are hierarchical:
Parent-child relationship between two nodes
basic operations of a tree:
tree traversals
insert node
delete node
searching

3. Level E R T L P M A S 1 2 3 root Child (of root)
Leaves or terminal nodes
Child (of root)
Depth of T: 2
Tree height: 4 1 2 3

4. Binary Trees
A tree in which no node can have more than two children

5. A General Tree & A Binary Tree

6. Balanced Binary Trees
A binary tree is balanced if the heights of any node’s two subtrees differ by no more than 1
Complete binary trees are balanced
Full binary trees are complete and balanced
B = HL - HR

7. Example: A A B A A B B C D E Balance Balance Balance Balance 2-1=1
1-1=0
3-2=1
1-2=-1 A A B
Balance
Balance A A B B C D E
Balance
Balance

8. + 3 2 Example of tree application: Represent algebraic formulas
Q: Write the following operations as binary tree and determine the (tree Depth, tree height, number of levels).
A. 2+3 + 1
Tree depth 2
Tree height
Number of levels 2 3

9. * / - 6 2 6 2 B. (6/2) * (20-4) 2 Tree depth 3 Tree height
Number of levels
B. (6/2) * (20-4) * / - 6 2 6 2

10. Tree traversal Type of Traversal Inorder traversal Preorder traversal
Postorder traversal

11. Inorder traversal
Recursively print out all data in the left subtree
Print the data at the root
Recursively print out all data in the right subtree
Pre-order traversal
Print the data at the root
Recursively print out all data in the left subtree
Recursively print out all data in the right subtree
Postorder traversal
Recursively print out all data in the left subtree
Recursively print out all data in the right subtree
Print the data at the root

12. Traverse the following binary tree using the three types of tree traversal6
Preorder (NLR)
6, 2, 1, 4, 3, 7, 10 , 9, 11
b. Postorder (LRN)
1, 3, 4, 2, 9, 11 ,10, 7, 6
c. Inorder (LNR)
1, 2, 3, 4, 6, 7, 9, 10, 11 7 2 1 4 10 3 9 11