1. 1 Binary Search Trees (BST) What is a Binary search tree? Why Binary search trees? Binary search tree implementation Insertion in a BST Deletion from a BST

2. 2 Binary Search Trees (Definition) A binary search tree (BST) is a binary tree that is empty or that satisfies the BST ordering property: 1.The key of each node is greater than each key in the left subtree, if any, of the node. 2.The key of each node is less than each key in the right subtree, if any, of the node. Thus, each key in a BST is unique. Examples: 6 8 2 41 79 53 AA B C D

3. 3 Why BST? DeletionInsertionRetrievalData Structure O(log n) FAST O(log n) FAST O(log n) FAST BST O(n) SLOW O(n) SLOW O(log n) FAST* Sorted Array O(n) SLOW O(n) SLOW O(n) SLOW Sorted Linked List BSTs provide good logarithmic time performance in the best and average cases. Average case complexities of using linear data structures compared to BSTs: *using binary search

4. 4 Binary Search Tree Implementation The BinarySearchTree class inherits the instance variables key, left, and right of the BinaryTree class: public class BinarySearchTree extends BinaryTree implements SearchableContainer { private BinarySearchTree getLeftBST(){ return (BinarySearchTree) getLeft( ) ; } private BinarySearchTree getRightBST( ){ return (BinarySearchTree) getRight( ) ; } //... }

5. 5 Binary Search Tree Implementation (Cont’d) The find method of the BinarySearchTree class: public Comparable find(Comparable comparable) { if(isEmpty()) return null; Comparable key = (Comparable) getKey(); if(comparable.compareTo(key)==0) return key; else if (comparable.compareTo(key)<0) return getLeftBST().find(comparable); else return getRightBST().find(comparable); }

6. 6 Binary Search Tree Implementation (Cont’d) The findMin method of the BinarySearchTree class: By the BST ordering property, the minimum key is the key of the left-most node that has an empty left-subtree. public Comparable findMin() { if(isEmpty()) return null; if(getLeftBST().isEmpty()) return (Comparable)getKey(); else return getLeftBST().findMin(); }

7. 7 Binary Search Tree Implementation (Cont’d) The findMax method of the BinarySearchTree class: By the BST ordering property, the maximum key is the key of the right-most node that has an empty right-subtree. 20 30 10 154 25 40 35 32 7 9 public Comparable findMax() { if(isEmpty()) return null; if(getRightBST().isEmpty()) return (Comparable)getKey(); else return getRightBST().findMax(); }

8. 8 Insertion in a BST By the BST ordering property, a new node is always inserted as a leaf node. The insert method, given in the next page, recursively finds an appropriate empty subtree to insert the new key. It then transforms this empty subtree into a leaf node by invoking the attachKey method: public void attachKey(Object obj) { if(!isEmpty()) throw new InvalidOperationException(); else { key = obj; left = new BinarySearchTree(); right = new BinarySearchTree(); }

9. 9 Insertion in a BST 1 6 8 2 4 79 3 56 8 2 4 79 3 5 6 8 2 4 79 3 5 6 8 2 4 79 3 5 11 1 public void insert(Comparable comparable){ if(isEmpty()) attachKey(comparable); else { Comparable key = (Comparable) getKey(); if(comparable.compareTo(key)==0) throw new IllegalArgumentException("duplicate key"); else if (comparable.compareTo(key)<0) getLeftBST().insert(comparable); else getRightBST().insert(comparable); }

10. 10 Deletion in a BST There are three cases: 1.The node to be deleted is a leaf node. 2.The node to be deleted has one non-empty child. 3.The node to be deleted has two non-empty children. CASE 1: DELETING A LEAF NODE Convert the leaf node into an empty tree by using the detachKey method: // In Binary Tree class public Object detachKey( ){ if(! isLeaf( )) throw new InvalidOperationException( ) ; else { Object obj = key ; key = null ; left = null ; right = null ; return obj ; }

11. 11 Deleting a leaf node (cont’d) Example: Delete 5 in the tree below: 7 15 2 41 840 63 9 5 Delete 5 7 15 2 41 840 63 9

12. 12 Deleting a one-child node CASE 2: THE NODE TO BE DELETED HAS ONE NON-EMPTY CHILD (a) The right subtree of the node x to be deleted is empty. Example: 20 35 5 8 3 2240 25 Delete 10 20 35 10 8 5 2240 3 25 6 target temp 6 target // Let target be a reference to the node x. BinarySearchTree temp = target.getLeftBST(); target.key = temp.key; target.left = temp.left; target.right = temp.right; temp = null;

13. 13 Deleting a one-child node (cont’d) (b) The left subtree of the node x to be deleted is empty. Example: Delete 8 7 15 2 41 840 63 12 5 target temp 149 7 15 2 41 1240 63 5 target 14 9 // Let target be a reference to the node x. BinarySearchTree temp = target.getRightBST(); target.key = temp.key; target.left = temp.left; target.right = temp.right; temp = null;

14. 14 CASE 3: DELETING A NODE THAT HAS TWO NON-EMPTY CHILDREN DELETION BY COPYING: METHOD#1 Copy the minimum key in the right subtree of x to the node x, then delete the one-child or leaf-node with this minimum key. Example: 7 15 2 41 840 63 9 5 Delete 7 8 15 2 41 940 63 5

15. 15 DELETING A NODE THAT HAS TWO NON-EMPTY CHILDREN DELETION BY COPYING: METHOD#2 Copy the maximum key in the left subtree of x to the node x, then delete the one-child or leaf-node with this maximum key. Example: 7 15 2 41 840 63 9 5 Delete 7 6 15 2 41 840 53 9

16. 16 Two-child deletion method#1 code // find the minimum key in the right subtree of the target node Comparable min = target.getRightBST().findMin(); // copy the minimum value to the target target.key = min; // delete the one-child or leaf node having the min target.getRightBST().withdraw(min); All the different cases for deleting a node are handled in the withdraw (Comparable key) method of BinarySearchTree class