O(lg n) Search Tree

1. O(lg n) Search Tree Tree T is a search tree made up of n elements: x0 x1 x2 x3 … xn-1 No function (except transverse) takes more than O(lg n) in the worst case.   Functions: createEmptyTree() returns a newly created empty binary tree delete(T, p) removes the node pointed to by p from the tree T insert(T, x) returns T with x added in the proper location search(T, key) returns a pointer to the node in T that has a key that matches key returns null if x is not found traverse(T) prints the contents of T in order isEmptyTree(T) returns true if T is empty and false if it is not

2. Homework 5 • Describe how to implement a search tree that has a worst time search, insert, and delete time of no more than O(lg n). This tree should have no number of element limit. • Do the six Search Tree functions. • Discuss how you would implement this if there was a maximum to the number of elements.

3. AVL Tree 0 54 0 +1 21 72 +1 -1 0 5 30 60 84 10 25 79 86

4. AVL Tree • The five functions are the same. • Except that the tree needs to be rebalanced after insertion or deletion. • Keep track of the path used to insert/delete. • Balance starting at the parent of the leaf inserted or deleted. • Work up to the root.

5. Insertion Case 1 0

6. Insertion Case 1 -1

7. Insertion Case 2 +1

8. Insertion Case 2 0

9. Insertion Case 3 +1

10. Insertion Case 3 +2

11. Deletion Case 1 -1

12. Deletion Case 1 0

13. Deletion Case 2 0

14. Deletion Case 2 +1

15. Deletion Case 3 +1

16. Deletion Case 3 +2

17. Single Rotation +2 +1

18. Double Rotation +2 -1

19. Single Rotation +2 +1

20. Single Rotation A +2 B +1

21. Single Rotation B A

22. Single Rotation B A +2 +1

23. Single Rotation B A 0 0

24. Single Rotation +2 +1

25. Single Rotation 0 +1

26. Single Rotation +1 0

27. Single Rotation +1 0

28. Single Rotation +1 0

29. Single Rotation 0 0

30. Double Rotation +2 -1

31. Double Rotation +1 0 0

32. Double Rotation1 A +2 C -1 B -1

33. Double Rotation1 B C A +2 -1 -1

34. Double Rotation1 A +2 C -1 B -1

35. Double Rotation1 A +2 C -1 B -1

36. Double Rotation1 A +1 C +1 B -1

37. Double Rotation1 A +1 B C +1 -1

38. Double Rotation1 A +1 B C +1 -1

39. Double Rotation1 A +1 B -1 C +1

40. Double Rotation1 A +2 B +1 C +1

41. Double Rotation1 A 0 B +1 C +1

42. Double Rotation1 A B 0 +1 C +1

43. Double Rotation1 B +1 A 0 C +1

44. Double Rotation1 B 0 A 0 C +1

45. Double Rotation1 B 0 A C 0 +1

46. Double Rotation2 A +2 C -1 B +1

47. Double Rotation2 A +2 C -1 B +1

48. Double Rotation2 A +2 C 0 B +1

49. Double Rotation2 A +2 B C 0 +1

50. Double Rotation2 A +2 B +2 C 0