Binary Search Trees

1. Binary Search Trees Lecture 6 Asst. Prof. Dr. İlker Kocabaş

2. Binary search tree sort

3. Binary-search-tree property

4. Binary-search-tree property • Binary Search Tree can be implemented as a linked data structure in which each node is an object with three pointer fields. • The three pointer fields left, right and p point to the nodes corresponding to the left child, right child and the parent respectively. • NIL in any pointer field signifies that there exists no corresponding child or parent. • The root node is the only node in the BTS structure with NIL in its p field.

5. Binary-search-tree property

6. Inorder-tree walk During this type of walk, we visit the root of a subtree between the left subtree visit and right subtree visit.

7. Preorder-tree walk In this case we visit the root node before the nodes in either subtree.

8. Postorder-tree walk We visit the root node after the nodes in its subtrees.

9. 1 2 6 Sorting by binary-search-tree 1 2 3 1 2 2 6 1 NIL • PRINT 2 4 7 3 NIL 3 PRINT 3 4 5 2 8 • 1 • NIL • PRINT 5 • NIL 5 3 PRINT 6 4 • 1 • NIL • PRINT 7 • 4 7 7 • 1 • NIL • PRINT 8 • NIL 8

10. Sorting by binary-search-tree

11. Sorting by binary-search-tree(contnd.)

12. Searching for a key 15 18 6 7 3 17 20 4 2 Search for 13 13 9

13. Searching for a key

14. Searching for minimum 15 18 6 7 3 17 20 4 2 13 Minimum 9

15. Searching for minimum

16. Searching for maximum 15 18 6 7 3 17 20 Maximum 4 2 13 9

17. Searching for maximum

18. Searching for successor The successor of a node x is the node with the smallest key greather than key[x]

19. Searching for successor 15 18 6 7 3 17 20 Successor of 15 4 2 13 9 Case 1: The right subtree of node xis nonempty

20. Searching for successor Successor of 13 15 18 6 7 3 17 20 4 2 13 9 Case 2: The right subtree of node xis empty

21. Searching for successor 15 Successor of 4 search until y.left = x 18 6 7 3 17 20 4 2 13 9 Case 2: The right subtree of node xis empty

22. Searching for successor  Go upper until y.left = x

23. Insertion 12 18 5 9 2 15 19 13 17 Inserting an item with key 13

24. Insertion

25. Deletion 15 16 5 5 12 3 2 20 13 z 10 23 18 6 7 Deleting an item with key 13 (z has no children)

26. Deletion 15 16 5 5 12 3 2 20 10 23 18 6 7 Deleting an item with key 13 (z has no children)

27. Deletion 15 16 z 5 5 12 3 2 20 13 10 23 18 6 7 Deleting a node with key 16 (z has only one child )

28. Deletion 15 5 5 20 12 3 2 23 18 13 10 6 7 Deleting a node with key 16 (z has only one child )

29. Deletion 15 z 5 16 5 12 3 2 20 10 13 18 23 6 y 7 Deleting a node with key 5 (6 successor of 5)(z has two children)

30. Deletion 15 6 y z 5 16 5 12 3 2 20 10 23 18 7 Deleting a node with key 5 (z has two children)

31. Deletion 15 6 16 5 12 3 2 20 10 23 18 7 Deleting a node with key 5 (z has two children)

32. Deletion

33. Analysis of BST sort The expected time to built the tree is asymptotically the same as the running time of quicksort

34. Red Black Trees

35. Balanced search trees

36. Red-black trees

37. Example of a red-black tree

38. Example of a red-black tree

39. Example of a red-black tree

40. Example of a red-black tree

41. Example of a red-black tree

42. Height of a red-black tree

43. Height of a red-black tree

44. Height of a red-black tree

45. Height of a red-black tree

46. Height of a red-black tree

47. Height of a red-black tree

48. Proof (continued)

49. Query operations

50. Modifying operations