Tutorial # March 2016 Aniruddha Chattoraj CPSC 319. Data Structures, Algorithms, and Their Applications Tutorial 01/03 Winter 17

Transcription

1 Tutorial # March 2016 Aniruddha Chattoraj CPSC 319 Data Structures, Algorithms, and Their Applications Tutorial 01/03 Winter 17

2 Trees A tree is a hierarchical data structure A binary tree is a tree data structure in which each node has at most two child nodes Full Tree and Complete Tree? Binary Search Tree: A BST is a binary tree where nodes are ordered in the following way: Each node contains one key (also known as data) The keys in the left sub-tree are less than the key in its parent node, in short L < P; The keys in the right sub-tree are greater than the key in its parent node, in short P < R; Duplicate keys are not allowed

3 BST Insertion Steps Search for a place to put a new element Insert the new element to this place Algorithm Starting from the root: Check, whether value in current node and a new value are equal. If so, duplicate is found. Otherwise, If a new value is less than the node's value: o If a current node has no left child, place for insertion has been found; o Otherwise, handle the left child with the same algorithm. If a new value is greater than the node's value: o If a current node has no right child, place for insertion has been found; o Otherwise, handle the right child with the same algorithm

4 BST Insertion Example Insert 4 to this tree

5 BST Insertion - Exercise Given a sequence of numbers: 11, 6, 8, 19, 4, 10, 5, 17, 43, 49, 31 Draw a binary search tree by inserting the above numbers from left to right Using the template code for a BST, implement the insert method

6 BST Insertion - Code Code class BSTNode { protected int value; protected BSTNode left, right;... public boolean insert(int value) { if (value == this.value) if (value < this.value) { if (left == null) { left = new BSTNode(value); return true; return left.insert(value); if (value > this.value) { if (right == null) { right = new BSTNode(value); return true; return right.insert(value); Code class IntBST { protected BSTNode root;... public boolean insert(int value) { if (root == null) { root = new BSTNode(value); return true; return root.insert(value);

7 BST - Search Stopping Criteria A node with necessary value is found Algorithm has no way to go Algorithm Starting from the root: Check, whether value in current node and searched value are equal. If so, value is found. Otherwise, If searched value is less than the node's value: If current node has no left child, searched value doesn't exist in the BST Otherwise, handle the left child with the same algorithm If the searched value is greater than the node's value: If current node has no right child, searched value doesn't exist in the BST Otherwise, handle the right child with the same algorithm

8 BST Search Example Search for 3 in the given tree

9 BST Search - Code Code class BSTNode { protected int value; protected BSTNode left, right;... public boolean search(int value) { if (value == this.value) return true; if (value < this.value) { if (left == null) return left.search(value); if (value > this.value) { if (right == null) return right.search(value); Code class IntBST { protected BSTNode root;... public boolean search(int value) { if (root == null) return root.search(value);

10 BST - Deletion Steps: Search for a node to remove If the node is found, run remove algorithm Cases Node to be removed has no children

11 BST Deletion Node to be removed has one child The node is cut from the tree and the algorithm links the single child (with its sub-tree) directly to the parent of the removed node.

12 BST Deletion Node to be removed has two children Use the principle of BST reorganization Example Choose minimum element from the right sub-tree (19 in the example) Replace 5 by 19 Hang 5 as a left child

13 BST Deletion Node to be removed has two children Use the principle of BST reorganization Example Choose minimum element from the right sub-tree (19 in the example) Replace 5 by 19 Hang 5 as a left child

14 BST Deletion Steps: Find a minimum value in the right sub-tree Replace value of the node to be removed with found minimum. Now, right sub-tree contains a duplicate! Apply remove to the right sub-tree to remove a duplicate Example remove 12 from the following BST Find minimum element in the right subtree of the node to be removed. In current example it is 19 Replace 12 with 19. Notice, that only values are replaced, not nodes. Now we have two nodes with the same value Remove 19 from the left sub-tree

15 BST Deletion Special Case node to be deleted is the root Dummy Root Approach: Create an auxiliary root Set the real root as the left child of the auxiliary root (i.e. the auxiliary root becomes the parent) Apply the remove algorithm After removal, set the (new) left child of the auxiliary root as the actual root

16 BST Deletion - Code Code class BSTNode { protected int value; protected BSTNode left, right;... public boolean remove(int value, BSTNode parent) { if (value < this.value) { if (left!= null) return left.remove(value, this); if (value > this.value) { if (right!= null) return right.remove(value, this); { if (left!= null && right!= null) { this.value = right.minvalue(); right.remove(this.value, this); if (parent.left == this) { parent.left = (left!= null)? left : right; if (parent.right == this) { parent.right = (left!= null)? left : right; return true; Code class IntBST { protected BSTNode root;... public boolean remove(int value) { if (root == null) { if (root.getvalue() == value) { BSTNode auxroot = new BSTNode(0); auxroot.setleftchild(root); boolean result = root.remove(value, auxroot); root = auxroot.getleft(); return result; { return root.remove(value, null); public int minvalue() { if (left == null) return value; return left.minvalue();