1. Binary Search Trees

2. Binary Search Trees Binary Tree labels at nodes
Labels have some order (comparable/comparator)
If a node has label x, then
every label in the right sub-tree is > x, and
every label in the left sub-tree is < x
This supports the dictionary ADT
Lookup, insert, delete
Running time = O(log n) (height of tree)

3. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59

4. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50

5. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 75

6. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75

7. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25

8. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61

9. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55

10. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55 30

11. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55 30 15

12. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55 68 30 15

13. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55 68 30 15 32

14. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55 68 30 15 32 36

15. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55 68 30 15 32 28 36

16. Example BST insertion
50, 75, 37, 25, 61, 55, 30, 15, 68, 32, 36, 28, 59 50 37 75 25 61 55 68 30 15 32 59 28 36

17. BST
An inOrder traversal of a BST accesses the elements in increasing order

18. Inorder public void printInOrder() { if (left != null)
left.printInOrder(); System.out.println(element);
if (right != null)
right.printInOrder(); }

19. Example BST 50 37 75 25 61 55 68 30 15 32 59 28 36

20. Example BST 50 37 75 25 61 55 68 30 15 32 59 28 36

21. Example BST 50 37 75 25 61 55 68 30 15 32 59 28 36

22. Example BST 50 37 75 25 61 55 68 30 15 32 59 28 36

23. Example BST 50 37 75 25 61 55 68 30 15 32 59 28 36

24. Example BST 15 50 37 75 25 61 55 68 30 15 32 59 28 36

25. Example BST 15 50 37 75 25 61 55 68 30 15 32 59 28 36

26. Example BST
15, 25 50 37 75 25 61 55 68 30 15 32 59 28 36

27. Example BST
15, 25 50 37 75 25 61 55 68 30 15 32 59 28 36

28. Example BST
15, 25 50 37 75 25 61 55 68 30 15 32 59 28 36

29. Example BST
15, 25, 28 50 37 75 25 61 55 68 30 15 32 59 28 36

30. Example BST
15, 25, 28 50 37 75 25 61 55 68 30 15 32 59 28 36

31. Example BST
15, 25, 28, 30 50 37 75 25 61 55 68 30 15 32 59 28 36

32. Example BST
15, 25, 28, 30 50 37 75 25 61 55 68 30 15 32 59 28 36

33. Example BST
15, 25, 28, 30, 32 50 37 75 25 61 55 68 30 15 32 59 28 36

34. Example BST
15, 25, 28, 30, 32 50 37 75 25 61 55 68 30 15 32 59 28 36

35. Example BST
15, 25, 28, 30, 32 50 37 75 25 61 55 68 30 15 32 59 28 36

36. Example BST
15, 25, 28, 30, 32 50 37 75 25 61 55 68 30 15 32 59 28 36

37. Example BST
15, 25, 28, 30, 32 50 37 75 25 61 55 68 30 15 32 59 28 36

38. Example BST
15, 25, 28, 30, 32 50 37 75 25 61 55 68 30 15 32 59 28 36

39. Example BST
15, 25, 28, 30, 32 50 37 75 25 61 55 68 30 15 32 59 28 36

40. Example BST
15, 25, 28, 30, 32, 37 50 37 75 25 61 55 68 30 15 32 59 28 36

41. Example BST
15, 25, 28, 30, 32, 37 50 37 75 25 61 55 68 30 15 32 59 28 36

42. Example BST
15, 25, 28, 30, 32, 37, 50 50 37 75 25 61 55 68 30 15 32 59 28 36

43. Example BST
15, 25, 28, 30, 32, 37, 50 50 37 75 25 61 55 68 30 15 32 59 28 36

44. Example BST
15, 25, 28, 30, 32, 37, 50 50 37 75 25 61 55 68 30 15 32 59 28 36

45. Example BST
15, 25, 28, 30, 32, 37, 50 50 37 75 25 61 55 68 30 15 32 59 28 36

46. Example BST
15, 25, 28, 30, 32, 37, 50, 55 50 37 75 25 61 55 68 30 15 32 59 28 36

47. Example BST
15, 25, 28, 30, 32, 37, 50, 55 50 37 75 25 61 55 68 30 15 32 59 28 36

48. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59 50 37 75 25 61 55 68 30 15 32 59 28 36

49. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59 50 37 75 25 61 55 68 30 15 32 59 28 36

50. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59 50 37 75 25 61 55 68 30 15 32 59 28 36

51. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61 50 37 75 25 61 55 68 30 15 32 59 28 36

52. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61 50 37 75 25 61 55 68 30 15 32 59 28 36

53. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61, 68 50 37 75 25 61 55 68 30 15 32 59 28 36

54. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61, 68 50 37 75 25 61 55 68 30 15 32 59 28 36

55. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61, 68 50 37 75 25 61 55 68 30 15 32 59 28 36

56. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61, 68, 75 50 37 75 25 61 55 68 30 15 32 59 28 36

57. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61, 68, 75 50 37 75 25 61 55 68 30 15 32 59 28 36

58. Example BST
15, 25, 28, 30, 32, 37, 50, 55, 59, 61, 68, 75 50 37 75 25 61 55 68 30 15 32 59 28 36

59. Lookup The label tells us which half of the tree to search
So, on average, we cut the size of the tree to search almost in half, which we can do log2n times
Basis:
If T is empty, fail; x not found
If T has label x at root, then found
Induction:
Let T have root label y
If x < y lookup x on left subtree
If x > y lookup x on right subtree

60. Insertion
Insertion function gets tree (reference to node) as an argument and returns a revised tree including inserted element
Basis:
If T is null create a new node with label x, return reference to the new node
Induction:
Let root of T have label y. If x < y, insert x into left subtree. The left-subtree refrence of root becomes whatever tree is returned. Same for x > y on right.

61. Deletion 50 37 75 25 61 55 68 30 15 32 59 28 36

62. Deletion: Leaf 50 37 75 25 61 55 68 30 15 32 59 28 36
Just remove it

63. Deletion: Leaf 50 37 75 25 61 55 68 30 15 32 59 28
Just remove it

64. Deletion: One child 50 37 75 25 61 55 68 30 15 32 59 28

65. Deletion: One child 50 37 75 25 61 55 68 30 15 32 59 28
Replace with other child

66. Deletion: One child 50 37 75 25 61 59 68 30 15 32 28
Replace with other child

67. Deletion: One child 50 37 61 25 59 68 30 15 32 28
Replace with other child

68. Deletion: Two children50 37 61 25 59 68 30 15 32 28

69. Deletion: Two children50 37 61 25 59 68 30 15
Replace element with leftmost
Value in rightmost tree 32 28

70. Deletion: Two children50 37 61 25 59 68 30 15
Replace element with leftmost
Value in rightmost tree 32 28
This element is the “successor”
In an inOrder traversal

71. Deletion: Two children59 37 61 25 68 30 15
Replace element with leftmost
Value in rightmost tree 32 28

72. Deletion Basis: If T is empty, just return T.
if x at root, delete root and fix up tree
return the fixed up tree
Induction:
If T has label y at root, delete x from left|right subtree if x<y|x>y. Replace left|right reference by returned tree. Return root.

73. Fixup (Deletemin)
If we need to delete the root of T, if it has one null subtree, just return the other subtree
Otherwise, find the least element in the right subtree (traverse leftmost path) move it to root.

74. Node class private static class Node { Object element;
Node left = null, right = null, parent;
Node (Object element, Node parent) {
this.element = element;
this.parent = parent; }

75. Example
root
element
left
right
parent
size
Eric  1

76. Example root element left right parent size Eric  4 Allen  Soumya 
Jack 

77. Runtime depends on the tree
50, 10, 20, 30, 25
25, 20, 30, 10, 50 50 10 20 30 25 25 20 30 50 10
O(log n)
O(n)

78. Deletion: Two children80 15
110 90
105

79. Deletion: Two childrenp 80 15
110 90 s
105

80. Deletion: Two childrenp 90 15
110 90 s
105

81. Deletion: Two childrenp 90
Delete -> 15
110 90 s
105

82. Deletion: Two childrenp 90
Delete -> 15
110 90
105
replacement

83. Deletion: Two children90 15
110
105