1. AVL Trees (Adelson – Velskii – Landis) AVL tree: 20 30 187 1625 5 1719

2. AVL Trees Not AVL tree: 27 20 30 187 1625 5 1719

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

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

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

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

7. AVL Tree Rotations Single rotations: 14 15 Rotation restores AVL balance: 16

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

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

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

11. AVL Tree Rotations Single rotations: 13 15 16 AVL violation – need to rotate 12 14 11

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

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

14. AVL Tree Rotations Single rotations: 13 15 16 12 14 11 10 AVL violation – need to rotate

15. AVL Tree Rotations Single rotations: 13 15 16 12 14 11 10 Rotation type:

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

18. AVL Tree Rotations Double rotations: 13 15 16 11 14 10 AVL violation - rotate 1 2 12

19. AVL Tree Rotations Double rotations: 13 15 16 11 14 10 Rotation type: 1 2 12

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

21. AVL Tree Rotations Double rotations: AVL violation – rotate: 1 10 13 15 16 11 14 2 12 3

22. AVL Tree Rotations Double rotations: Rotation type: 1 10 13 15 16 11 14 2 12 3

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

24. AVL Tree Rotations Double rotations: AVL violation - rotate 1 3 13 15 16 10 14 2 11 12 4

25. AVL Tree Rotations Double rotations: Rotation type: 1 3 13 15 16 10 14 2 11 12 4

26. AVL Tree Rotations Double rotations: 10 13 15 2 11 1 3 4 121416 Now insert 5.

27. AVL Tree Rotations Double rotations: 10 13 15 2 11 1 3 4 121416 AVL violation – rotate. 5

28. AVL Tree Rotations Single rotations: 10 13 15 2 11 1 3 4 121416 5 Rotation type:

29. AVL Tree Rotations Single rotations: 10 13 15 2 11 1 4 5 121416 3 AVL balance restored: Now insert 7.

30. AVL Tree Rotations Single rotations: 10 13 15 2 11 1 4 5 121416 3 AVL violation – rotate. 7

31. AVL Tree Rotations Single rotations: 10 13 15 2 11 1 4 5 121416 3 7 Rotation type:

32. AVL Tree Rotations Double rotations: 10 13 15 4 11 2 5 7 121416 31 AVL balance restored. Now insert 6.

33. AVL Tree Rotations Double rotations: 10 13 15 4 11 2 5 7 121416 31 AVL violation - rotate. 6

34. AVL Tree Rotations Double rotations: 10 13 15 4 11 2 5 7 121416 31 Rotation type: 6

35. AVL Tree Rotations Double rotations: 10 13 15 4 11 2 6 7 121416 31 AVL balance restored. Now insert 9 and 8. 5

36. AVL Tree Rotations Double rotations: 10 13 15 4 11 2 6 7 121416 31 AVL violation - rotate. 9 8 5

37. AVL Tree Rotations Double rotations: 10 13 15 4 11 2 6 7 121416 31 Rotation type: 9 8 5

38. AVL Tree Rotations Final tree: 10 13 15 4 11 2 6 8 121416 31 Tree is almost perfectly balanced 5 97