1. Lecture No.20 Data Structures Dr. Sohail Aslam

2. AVL Tree AVL (Adelson-Velskii and Landis) tree.
An AVL tree is identical to a BST except
height of the left and right subtrees can differ by at most 1.
height of an empty tree is defined to be (–1).
Start of 20 (Nov 29)

3. AVL Tree
An AVL Tree
height 5 2 8 1 1 4 7 2 3 3

4. AVL Tree
Not an AVL tree
height 6 1 8 1 1 4 2 3 5 3

5. Balanced Binary Tree
The height of a binary tree is the maximum level of its leaves (also called the depth).
The balance of a node in a binary tree is defined as the height of its left subtree minus height of its right subtree.
Here, for example, is a balanced tree. Each node has an indicated balance of 1, 0, or –1.

6. Balanced Binary Tree -1 1 1 -1
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7. Insertions and effect on balance
Balanced Binary Tree
Insertions and effect on balance -1 1 1 -1 B B U1 U2 U3 U4 B B B B U5 U6 U7 U8 U9
U10
U11
U12

8. Balanced Binary Tree
Tree becomes unbalanced only if the newly inserted node
is a left descendant of a node that previously had a balance of 1 (U1 to U8),
or is a descendant of a node that previously had a balance of –1 (U9 to U12)

9. Insertions and effect on balance
Balanced Binary Tree
Insertions and effect on balance -1 1 1 -1 B B U1 U2 U3 U4 B B B B U5 U6 U7 U8 U9
U10
U11
U12

10. Balanced Binary Tree Consider the case of node that was previously 1-1 1 1 -1 B B U1 U2 U3 U4 B B B B U5 U6 U7 U8 U9
U10
U11
U12

11. Inserting New Node in AVL Tree1 B T1 T2 T3 1

12. Inserting New Node in AVL Tree2 B 1 T1 T2 T3 1 2
new

13. Inserting New Node in AVL Tree2 B A B 1 T1 T1 T2 T3 T2 1 T3 2
new
new
Inorder: T1 B T2 A T3
Inorder: T1 B T2 A T3

14. AVL Tree Building Example
Let us work through an example that inserts numbers in a balanced search tree.
We will check the balance after each insert and rebalance if necessary using rotations.

15. AVL Tree Building Example
Insert(1) 1

16. AVL Tree Building Example
Insert(2) 1 2

17. AVL Tree Building Example
Insert(3) single left rotation 1 -2 2 3

18. AVL Tree Building Example
Insert(3) single left rotation 1 -2 2 3

19. AVL Tree Building Example
Insert(3) 2 1 3
End of lecture 20