2. Binary Search Tree 황승원 Fall 2011 CSE, POSTECH

3. 2 2 Search Trees Search trees are ideal for implementing dictionaries – Similar or better performance than skip lists and hashing – Particularly ideal for accessing data sequentially or by rank Types of search trees – Binary search trees – AVL trees – Red-black trees – B-trees

4. 3 3 Binary Search Tree Definition A binary tree that may be empty. A nonempty binary search tree satisfies the following properties 1. Each node has a (key, value) pair and no two nodes have the same key (i.e., all keys are distinct). 2. For every node x, all keys in the left subtree of x are smaller than that in x. 3. For every node x, all keys in the right subtree of x are larger than that in x. 4. The left and right subtrees of the root are also binary search trees

5. 4 Binary Search Trees Dictionary Operations:  get(key)  put(key, value)  remove(key) Additional operations:  ascend()  Elements in the ascending order  get(rank) (indexed binary search tree)  Get the element in the rank th order  remove(rank) (indexed binary search tree)  Remove the element in the rank th order

6. 5 Complexity Of Dictionary Operations get(), put() and remove() n is number of elements in dictionary

7. 6 Definition Of Binary Search Tree A binary tree. Each node has a (key, value) pair. For every node x, all keys in the left subtree of x are smaller than that in x. For every node x, all keys in the right subtree of x are greater than that in x.

8. 7 Example Binary Search Tree 20 10 6 28 15 40 30 25 Only keys are shown.

9. 8 The Operation ascend() 20 10 6 28 15 40 30 25 Do an inorder traversal. O(n) time.

10. 9 The Operation get() 20 10 6 28 15 40 30 25 Complexity is O(height) = O(n), where n is number of nodes/elements.

11. 10 The Operation put() 20 10 6 28 15 40 30 25 Put a pair whose key is 35. 35

12. 11 The Operation put() Put a pair whose key is 7. 20 10 6 28 15 40 30 2535 7

13. 12 The Operation put() 20 10 6 28 15 40 30 25 Put a pair whose key is 18. 35 7 18

14. 13 The Operation put() 20 10 6 28 15 40 30 25 Complexity of put() is O(height). 35 7 18

15. 14 The Operation remove() Three cases:  Element is in a leaf.  Element is in a degree 1 node.  Element is in a degree 2 node.

16. 15 Remove From A Leaf Remove a leaf element. key = 7 20 10 6 28 15 40 30 2535 7 18

17. 16 Remove From A Leaf (contd.) Remove a leaf element. key = 35 20 10 6 28 15 40 30 2535 7 18

18. 17 Remove From A Degree 1 Node Remove from a degree 1 node. key = 40 20 10 6 28 15 40 30 2535 7 18

19. Remove From A Degree 1 Node (contd.) Remove from a degree 1 node. key = 15 20 10 6 28 15 40 30 2535 7 18

20. 19 Remove From A Degree 2 Node Remove from a degree 2 node. key = 10 20 10 6 28 15 40 30 2535 7 18

21. 20 Remove From A Degree 2 Node 20 10 6 28 15 40 30 25 Replace with largest key in left subtree (or smallest in right subtree). 35 7 18

22. 21 Remove From A Degree 2 Node 20 10 6 28 15 40 30 25 Replace with largest key in left subtree (or smallest in right subtree). 35 7 18

23. 22 Remove From A Degree 2 Node 20 8 6 28 15 40 30 25 Replace with largest key in left subtree (or smallest in right subtree). 35 7 18

24. 23 Remove From A Degree 2 Node 20 8 6 28 15 40 30 25 Largest key must be in a leaf or degree 1 node. 35 7 18

25. 24 Another Remove From A Degree 2 Node Remove from a degree 2 node. key = 20 20 10 6 28 15 40 30 2535 7 18

26. 25 Remove From A Degree 2 Node 20 10 6 28 15 40 30 25 Replace with largest in left subtree. 35 7 18

27. 26 Remove From A Degree 2 Node 20 10 6 28 15 40 30 25 Replace with largest in left subtree. 35 7 18

28. 27 Remove From A Degree 2 Node 18 10 6 28 15 40 30 25 Replace with largest in left subtree. 35 7 18

29. 28 Remove From A Degree 2 Node 18 10 6 28 15 40 30 25 Complexity is O(height). 35 7

30. 29 Binary Search Trees with Duplicates We can remove the requirement that all elements in a binary search tree need distinct keys How? – Replace “smaller” in property 2 by “smaller or equal to” – Replace “larger” in property 3 by “larger or equal to” Then binary search trees can have duplicate keys

31. 30 Indexed Binary Search Tree Binary search tree. Each node has an additional field.  leftSize = number of nodes in its left subtree

32. 31 Example Indexed Binary Search Tree 20 10 6 28 15 40 30 2535 7 18 0 0 1 1 4 0 0 7 0 0 1 3 leftSize values are in red

33. 32 leftSize And Rank Rank of an element is its position in inorder (inorder = ascending key order). [2,6,7,8,10,15,18,20,25,30,35,40] rank(2) = 0 rank(15) = 5 rank(20) = 7 leftSize(x) = rank(x) with respect to elements in subtree rooted at x

34. 33 leftSize And Rank 20 10 6 28 15 40 30 2535 7 18 0 0 1 1 4 0 0 7 0 0 1 3 sorted list = [2,6,7,8,10,15,18,20,25,30,35,40]

35. 34 get(index) And remove(index) 7 20 10 6 28 15 40 30 2535 7 18 0 0 1 1 4 0 0 0 0 1 3 sorted list = [2,6,7,8,10,15,18,20,25,30,35,40]

36. 35 get(index) And remove(index) if index = x.leftSize desired element is x.element if index < x.leftSize desired element is index ’ th element in left subtree of x if index > x.leftSize desired element has index=(index - x.leftSize-1) in the right subtree of x

37. 36 Challenge O(height)=O(n) In which case? How to avoid this?

38. 37 Coming Up Next READING: Ch 15 NEXT: Balanced Binary Tree (Ch 16.1~2)