Binary Search Trees

1. Binary Search Trees CSE, POSTECH

2. Search Trees • Search trees are ideal for implementing dictionaries • Similar or better performance than skip lists and hash tables • Particularly ideal for accessing data sequentially or by rank • In this chapter, we will learn • Binary search trees • Indexed binary search trees

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

4. Figure 14.1 Binary Trees Examples of Binary Trees • Which of the above trees are binary search trees?  (b) and (c) • Why isn’t (a) a binary search tree?  It violates the property #3

5. Indexed Binary Search Trees Definition • Binary search tree. • Each node has an additional field ‘LeftSize’. • Support search and delete operations by rankas well as all the binary search tree operations. • LeftSize • the number of elements in its left subtree • the rank of an element with respect to the elements in its subtree (e.g., the fourth element in sorted order)

6. 7 20 4 3 10 40 1 0 1 6 15 30 0 1 0 0 0 2 8 18 25 35 0 7 Indexed Binary Search Tree Example • What is the Leftsize for each node? • LeftSize values are in red.

7. 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 the subtree rooted at x • See Figure 14.2 for more examples

8. Exercise • Do Exercise 14.1

9. The bsTree ADT • See ADT 14.1 for bsTree ADT • See ADT 14.2 for IndexedBSTree ADT • See Programs 14.1 and 14.2 for C++ abstract class definitions for bsTree and IndexedBSTree

10. The Class binarySearchTree • Since the number of elements in a binary search tree as well as its shape changes as operations are performed, a binary search tree is usually represented using the linked representationof Section 11.4.2 • We can define binarySearchTree as a derived class of linkedBinaryTree (Section 11.8)

11. 20 10 40 6 15 30 2 8 25 The Operation Ascend() • How can we output all elements in ascending order of keys? • Do an inorder traversal (left, root, right). • What would be the output? • 2, 6, 8, 10, 15, 20, 25, 30, 40

12. The Operation Search(key, e) • Search begins at the root • If the root is NULL, the search tree is empty and the search fails. • If key is less than the root, then left subtree is searched • If key is greater than the root, then right subtree is searched • If key equals the root, then the search terminates successfully • The time complexity for search is O(height) • See Program 11.4 for the search operation code

13. The Operation Insert(key, e) • To insert a new element e into a binary search tree, we must first verify that its key does not already exist by performing a search in the tree • If the search is successful, we do not insert • If the search is unsuccessful, then the element is inserted at the point the search terminated • Why insert it at that point? • The time complexity for insert is O(height) • See Figure 14.3 for examples • See Program 14.5 for the insert operation code

14. 20 10 40 6 15 30 2 8 25 35 Insert Example We wish to insert an element with the key 35. Where should it be inserted?

15. 20 10 40 6 15 30 2 8 25 35 7 Insert Example Insert an element with the key 7.

16. 20 10 40 6 15 30 2 8 18 25 35 7 Insert Example Insert an element with the key 18.

17. The Operation Delete(key, e) • For deletion, there are three cases for the element to be deleted: • Element is in a leaf. • Element is in a degree 1 node (i.e., has exactly one nonempty subtree). • Element is in a degree 2 node (i.e., has exactly two nonempty subtrees).

18. 20 10 40 6 15 30 2 8 18 25 35 7 Case 1: Delete from a Leaf • For case 1, we can simply discard the leaf node. • Example, delete a leaf element. key=7

19. 20 10 40 6 15 30 2 8 18 25 35 Case 1: Delete from a Leaf Delete a leaf element. key=35

20. 20 10 40 6 15 30 2 8 18 25 Case 2: Delete from a Degree 1 Node • Which nodes have a degree 1? • Example: Delete key=40

21. 20 10 6 15 30 2 8 18 25 Case 2: Delete from a Degree 1 Node Delete from a degree 1 node. key=15

22. 20 10 40 6 15 30 2 8 18 25 35 7 Case 3: Delete from a Degree 2 Node • Which nodes have a degree 2? • Example: Delete key=10

23. 20 10 40 6 15 30 2 8 18 25 35 7 Case 3: Delete from a Degree 2 Node • Replace with the largest key in the left subtree(or the smallest in the right subtree) • Which node is the largest key in the left subtree?

24. 20 8 40 6 15 30 2 8 18 25 35 7 Case 3: Delete from a Degree 2 Node The largest key must be in a leaf or degree 1 node.

25. Case 3: Delete from a Degree 2 Node • Note that the node with largest key in the left subtree (as well as that with smallest in the right subtree) is guaranteed to be in a node with either zero or one nonempty subtree • How can we find the node with largest key in the left subtree of a node?  by moving to the root of that subtree and then following a sequence of right-child pointers until we reach a node whose right-child pointer is NULL • How can we find the node with smallest key in the right subtree of a node?  by moving to the root of that subtree and then following a sequence of left-child pointers until we reach a node whose left-child pointer is NULL

26. 20 10 40 6 15 30 2 8 18 25 35 7 Another Delete from a Degree 2 Node • Delete from a degree 2 node. key=20 • Replace with the largest in the left subtree.

27. 18 10 40 6 15 30 2 8 25 35 7 Another Delete from a Degree 2 Node The result is • The time complexity of delete is O(height). • See more delete examples in Figure 14.4 • See Program 14.6 for the delete operation code

28. 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

29. The Class dBSTree • The binary search tree with duplicates (dBSTree) • We can implement this by changing the while loop of binarySearchTree::Insert (Program 14.5) • See Program 14.7 for the new while loop for dBSTree

30. Indexed Binary Search Tree – Search & Delete • IndexSearch(rank) returns the rankth element • IndexDelete(rank) deletes the rankth element • If rank = x.LeftSize, desired element is x.element. • If rank < x.LeftSize, desired element is rankth element in left subtree of x. • If rank > x.LeftSize, desired element is(rank-(x.LeftSize+1))th element in right subtree of x. • Read Section 14.5

31. 7 20 4 3 10 40 1 0 1 6 15 30 0 1 0 0 0 2 8 18 25 35 0 7 Indexed Binary Search Example • What is the 4th element in this IndexedBST? • What is the 10th element in this IndexedBST?

32. READING • Do Exercise 14.7 • Read Sections 14.1-14.5