AVL Trees

1. Single Rotation to Fix Case 1(left-
left)
k2 violates
An insertion in subtree X,
AVL property violated at node k2
Solution: single rotation

2. Single Rotation to Fix Case 4 (right-
right)
• Case 4 is a symmetric case to case 1
• Insertion takes O(Height of AVL Tree)
time, Single rotation takes O(1) time
An insertion in subtree Z
k1 violates

3. Single Rotation Fails to fix Case 2&3
• Single rotation fails to fix case 2&3
• Take case 2 as an example (case 3 is a
symmetry to it )
– The problem is subtree Y is too deep
– Single rotation doesn’t make it any less deep
Single rotation result
Case 2: violation in k2 because of
insertion in subtree Y

4. Double Rotation to Fix Case 2 (left-right)
• Facts
– The new key is inserted in the subtree B or C
– The AVL-property is violated at k3
– k3-k1-k2 forms a zig-zag shape
• Solution
– We cannot leave k3 as the root
– The only alternative is to place k2 as the new
root
Double rotation to fix case 2

5. Double Rotation to fix Case 3(right-left)
• Facts
– The new key is inserted in the subtree B or C
– The AVL-property is violated at k1
– k2-k3-k2 forms a zig-zag shape
• Case 3 is a symmetric case to case 2
Double rotation to fix case 3

6. AVL Tree Example:
• Insert 14, 17, 11, 7, 53, 4, 13 into an empty AVL tree
14
17
11
7 53
4

7. AVL Tree Example:
• Insert 14, 17, 11, 7, 53, 4, 13 into an empty AVL tree
14
17
7
4 53
11
13

8. AVL Tree Example:
• Now insert 12 Right Left case – first rotate Right W.R.T 13 and
then Left rotate W.R.T 11
14
17
7
4 53
11
13
12

9. AVL Tree Example:
• Now insert 12
14
17
7
4 53
11
12
13

10. AVL Tree Example:
• Now the AVL tree is balanced.
14
17
7
4 53
12
13
11

11. AVL Tree Example:
• Now insert 8
14
17
7
4 53
12
13
11
8

12. AVL Tree Example:
• Now insert 8
14
17
7
4 53
11
12
8
13

13. AVL Tree Example:
• Now the AVL tree is balanced.
14
17
7
4
53
11
12
8 13

14. AVL Tree Example:
• Now remove 53
14
17
7
4
53
11
12
8 13

15. AVL Tree Example:
• Now remove 53, unbalanced
14
17
7
4
11
12
8 13

16. AVL Tree Example:
• Balanced! Remove 11
14
17
7
4
11
12
8
13

17. AVL Tree Example:
• Remove 11, replace it with the largest in its left branch
14
17
7
4
8
12
13

18. AVL Tree Example:
• Remove 8, unbalanced
14
17
4
7
12
13

19. AVL Tree Example:
• Remove 8, unbalanced
14
17
4
7
12
13

20. AVL Tree Example:
• Balanced!!
14
17
4
7
12
13

21. In Class Exercises
• Build an AVL tree with the following values:
15, 20, 24, 10, 13, 7, 30, 36, 25

22. 15
15, 20, 24, 10, 13, 7, 30, 36, 25
20
24
15
20
24
10
13
15
20
24
13
10
13
20
24
15
10

23. 13
20
24
15
10
15, 20, 24, 10, 13, 7, 30, 36, 25
7
13
20
24
15
10
7
30
36
13
20
30
15
10
7
36
24

24. 13
20
30
15
10
7
36
24
15, 20, 24, 10, 13, 7, 30, 36, 25
25
13
20
30
15
10
7
36
24
25
13
24
36
20
10
7
25
30
15

25. Remove 24 and 20 from the AVL tree.
13
24
36
20
10
7
25
30
15
13
20
36
15
10
7
25
30
13
15
36
10
7
25
30
13
30
36
10
7
25
15