1. Topic 10
Trees

2. Tree Definitions

3. Tree Definitions

4. Tree Definitions

5. Binary Search Tree

6. Traversal
Operations tree

7. Announcements Participation 4 is up Homework 3 grading

9. Search

11. Insertion

13. Deletion
Need
Min/max
Successor/predecessor

16. Pseudocode

17. Pseudocode

18. AVL

19. Height of AVL Tree

20. Proof

22. Insertion

23. Example

24. Single Right

25. Left-Right

26. Four cases

27. Deletions

28. Example

30. Announcements
Test in 2 weeks
Participation due tomorrow
Clarification

31. Splay Trees

32. Amortized Running Time
Runtime:

33. Basic Operations

34. When to Splay

35. Zig

36. Zig-Zag

37. Zig-zig

38. Example

43. Deletion

46. Optimal BST

47. Notation

48. Example

49. Example

50. More Notation
(i,j)
(j+1, j)

51. Observations

52. Observations

53. Announcements Participation due
Clarifications on previous topics today
Splay tree deletion
Universal Hashing

54. Guaranteeing O(n) time

55. Birthday Paradox

56. Universal Hashing (clarification)

59. Computing the Optimal BST
How to compute cost?
Subtree at different depths

60. w(i,j)

61. AST

62. AST

63. Example
Sequences/Probabilities

64. Leaves

65. The Big Idea

67. Subtrees with one key

68. Subtrees with 2 keys

69. Solution

70. Announcements Homework 5 due Homework 6 up
Solutions will be up later today
Homework 6 up

73. 2-3 Trees
A search tree
Properties

75. Insertion

77. Deletion
Remove
Borrow
Coalesce
Apply recursively

78. Remove

79. Borrow

80. Coalesce

81. Generalizing/Example

90. 2-3-4 trees, B-trees
Why use 2-3 trees?