1. AVL Trees (Adelson – Velskii – Landis)20 16 25 30 7 18 5 17 19

2. AVL Trees
Not AVL tree: 20 16 25 30 7 18 5 17 19 27

3. AVL Tree Rotations Single rotations: insert 14, 15, 16, 13, 12, 11, 10
First insert 14 and 15: 14 15
Now insert 16.

4. AVL Tree Rotations Single rotations:
Inserting 16 causes AVL violation: 14 15 16
Need to rotate.

5. AVL Tree Rotations Single rotations:
Inserting 16 causes AVL violation: 14 15 16
Need to rotate.

6. AVL Tree Rotations
Single rotations:
Rotation type: 14 15 16

7. AVL Tree Rotations Single rotations: Rotation restores AVL balance: 1514 16

8. AVL Tree Rotations Single rotations: Now insert 13 and 12:15 14 16 13 12
AVL violation - need to rotate.

9. AVL Tree Rotations
Single rotations:
Rotation type: 15 14 16 13 12

10. AVL Tree Rotations
Single rotations: 15 13 16 12 14
Now insert 11.

11. AVL Tree Rotations Single rotations: AVL violation – need to rotate 1513 16 12 14 11
AVL violation – need to rotate

12. AVL Tree Rotations
Single rotations:
Rotation type: 15 13 16 12 14 11

13. AVL Tree Rotations
Single rotations: 13 12 15 16 14 11
Now insert 10.

14. AVL Tree Rotations Single rotations: AVL violation – need to rotate 1312 15 16 14 11 10
AVL violation – need to rotate

15. AVL Tree Rotations Single rotations: Rotation type: 13 12 15 16 14 1110

16. AVL Tree Rotations Single rotations: AVL balance restored. 13 11 15 1012 14 16
AVL balance restored.

17. AVL Tree Rotations Double rotations: insert 1, 2, 3, 4, 5, 7, 6, 9, 8
First insert 1 and 2: 13 11 15 10 12 14 16

18. AVL Tree Rotations Double rotations: AVL violation - rotate 13 11 1516 14 10 12 1 2

19. AVL Tree Rotations Double rotations: Rotation type: 13 11 15 16 14 1012 1 2

20. AVL Tree Rotations Double rotations: AVL balance restored:13 11 15 2 12 14 16 1 10
Now insert 3.

21. AVL Tree Rotations Double rotations: AVL violation – rotate: 13 11 152 12 14 16 1 10 3

22. AVL Tree Rotations Double rotations: Rotation type: 13 11 15 2 12 1416 1 10 3

23. AVL Tree Rotations Double rotations: AVL balance restored:13 10 15 2 11 14 16 1 3 12
Now insert 4.

24. AVL Tree Rotations Double rotations: AVL violation - rotate 13 10 15 211 14 16 1 3 12 4

25. AVL Tree Rotations Double rotations: Rotation type: 13 10 15 2 11 1416 1 3 12 4

26. AVL Tree Rotations Double rotations: Now insert 5. 10 13 2 1 11 15 312 14 16 4
Now insert 5.

27. AVL Tree Rotations Double rotations: AVL violation – rotate. 10 13 215 1 11 3 12 14 16 4 5
AVL violation – rotate.

28. AVL Tree Rotations Single rotations: Rotation type: 10 13 2 15 1 11 312 14 16 4 5

29. AVL Tree Rotations Single rotations: AVL balance restored:10 13 2 15 1 11 4 12 14 16 3 5
Now insert 7.

30. AVL Tree Rotations Single rotations: AVL violation – rotate. 10 13 215 1 11 4 12 14 16 3 5 7

31. AVL Tree Rotations Single rotations: Rotation type: 10 13 2 15 1 11 412 14 16 3 5 7

32. AVL Tree Rotations Double rotations: AVL balance restored.10 13 4 11 15 2 5 12 14 16 7 1 3
Now insert 6.

33. AVL Tree Rotations Double rotations: AVL violation - rotate. 10 13 411 15 2 5 12 14 16 7 1 3 6

34. AVL Tree Rotations Double rotations: Rotation type: 10 13 4 11 15 2 512 14 16 7 1 3 6

35. AVL Tree Rotations Double rotations: AVL balance restored.10 13 4 11 15 2 6 12 14 16 1 3 5 7
Now insert 9 and 8.

36. AVL Tree Rotations Double rotations: AVL violation - rotate. 10 13 415 2 11 6 12 14 16 1 3 5 7 9 8

37. AVL Tree Rotations Double rotations: Rotation type: 10 13 4 15 2 11 612 14 16 1 3 5 7 9 8

38. AVL Tree Rotations Final tree: Tree is almost perfectly balanced 10 134 15 2 11 6 12 14 16 1 3 5 8 7 9