Binary Trees, Binary Search Trees RIZWAN REHMAN CENTRE FOR COMPUTER STUDIES DIBRUGARH UNIVERSITY

Binary Search Trees / Slide 2 Trees * Linear access time of linked lists is prohibitive n Does there exist any simple data structure for which the running time of most operations (search, insert, delete) is O(log N)?

Binary Search Trees / Slide 3 Trees * A tree is a collection of nodes n The collection can be empty n (recursive definition) If not empty, a tree consists of a distinguished node r (the root), and zero or more nonempty subtrees T 1, T 2,...., T k, each of whose roots are connected by a directed edge from r

Binary Search Trees / Slide 4 Some Terminologies * Child and parent n Every node except the root has one parent n A node can have an arbitrary number of children * Leaves n Nodes with no children * Sibling n nodes with same parent

Binary Search Trees / Slide 5 Some Terminologies * Path * Length n number of edges on the path * Depth of a node n length of the unique path from the root to that node n The depth of a tree is equal to the depth of the deepest leaf * Height of a node n length of the longest path from that node to a leaf n all leaves are at height 0 n The height of a tree is equal to the height of the root * Ancestor and descendant n Proper ancestor and proper descendant

Binary Search Trees / Slide 6 Example: UNIX Directory

Binary Search Trees / Slide 7 Binary Trees * A tree in which no node can have more than two children * The depth of an “average” binary tree is considerably smaller than N, eventhough in the worst case, the depth can be as large as N – 1.

Binary Search Trees / Slide 8 Example: Expression Trees * Leaves are operands (constants or variables) * The other nodes (internal nodes) contain operators * Will not be a binary tree if some operators are not binary

Binary Search Trees / Slide 9 Tree traversal * Used to print out the data in a tree in a certain order * Pre-order traversal n Print the data at the root n Recursively print out all data in the left subtree n Recursively print out all data in the right subtree

Binary Search Trees / Slide 10 Preorder, Postorder and Inorder * Preorder traversal n node, left, right n prefix expression  ++a*bc*+*defg

Binary Search Trees / Slide 11 Preorder, Postorder and Inorder * Postorder traversal n left, right, node n postfix expression  abc*+de*f+g*+ * Inorder traversal n left, node, right. n infix expression  a+b*c+d*e+f*g

Binary Search Trees / Slide 12 * Preorder

Binary Search Trees / Slide 13 * Postorder

Binary Search Trees / Slide 14 Preorder, Postorder and Inorder

Binary Search Trees / Slide 15 Binary Trees * Possible operations on the Binary Tree ADT n parent n left_child, right_child n sibling n root, etc * Implementation n Because a binary tree has at most two children, we can keep direct pointers to them

Binary Search Trees / Slide 16 compare: Implementation of a general tree

Binary Search Trees / Slide 17 Binary Search Trees * Stores keys in the nodes in a way so that searching, insertion and deletion can be done efficiently. Binary search tree property n For every node X, all the keys in its left subtree are smaller than the key value in X, and all the keys in its right subtree are larger than the key value in X

Binary Search Trees / Slide 18 Binary Search Trees A binary search tree Not a binary search tree

Binary Search Trees / Slide 19 Binary search trees * Average depth of a node is O(log N); maximum depth of a node is O(N) Two binary search trees representing the same set:

Binary Search Trees / Slide 20 Implementation

Binary Search Trees / Slide 21 Searching BST * If we are searching for 15, then we are done. * If we are searching for a key < 15, then we should search in the left subtree. * If we are searching for a key > 15, then we should search in the right subtree.

Binary Search Trees / Slide 22

Binary Search Trees / Slide 23 Searching (Find) * Find X: return a pointer to the node that has key X, or NULL if there is no such node * Time complexity n O(height of the tree)

Binary Search Trees / Slide 24 Inorder traversal of BST * Print out all the keys in sorted order Inorder: 2, 3, 4, 6, 7, 9, 13, 15, 17, 18, 20

Binary Search Trees / Slide 25 findMin/ findMax * Return the node containing the smallest element in the tree * Start at the root and go left as long as there is a left child. The stopping point is the smallest element * Similarly for findMax * Time complexity = O(height of the tree)

Binary Search Trees / Slide 26 insert * Proceed down the tree as you would with a find * If X is found, do nothing (or update something) * Otherwise, insert X at the last spot on the path traversed * Time complexity = O(height of the tree)

Binary Search Trees / Slide 27 delete * When we delete a node, we need to consider how we take care of the children of the deleted node. n This has to be done such that the property of the search tree is maintained.

Binary Search Trees / Slide 28 delete Three cases: (1) the node is a leaf n Delete it immediately (2) the node has one child n Adjust a pointer from the parent to bypass that node

Binary Search Trees / Slide 29 delete (3) the node has 2 children n replace the key of that node with the minimum element at the right subtree n delete the minimum element  Has either no child or only right child because if it has a left child, that left child would be smaller and would have been chosen. So invoke case 1 or 2. * Time complexity = O(height of the tree)

Binary Search Trees / Slide 30 Extended Binary Tree A binary tree in which special nodes are added wherever a nullbinary tree subtree was present in the original tree so that each node in the original tree (except the root node) has degree three (Knuth 1997, p. 399).

Binary Search Trees / Slide 31 Binary Search Trees * A binary tree: n No node has more than two child nodes (called child subtrees). n Child subtrees must be differentiated, into:  Left-child subtree  Right-child subtree * A search tree: n For every node, p:  All nodes in the left subtree are < p  All nodes in the right subtree are > p

Binary Search Trees / Slide 32 Binary Search Tree - Example

Binary Search Trees / Slide 33 Binary Search Trees (cont) * Searching for a value is in a tree of N nodes is: n O(log N) if the tree is “balanced” n O(N) if the tree is “unbalanced”

Binary Search Trees / Slide 34 “Unbalanced” Binary Search Trees * Below is a binary search tree that is NOT “balanced”

Binary Search Trees / Slide 35 Properties of Binary Trees * A binary tree is a full binary tree if and only if: n Each non leaf node has exactly two child nodes n All leaf nodes have identical path length * It is called full since all possible node slots are occupied

Binary Search Trees / Slide 36 A Full Binary Tree - Example

Binary Search Trees / Slide 37 Full Binary Trees * A Full binary tree of height h will have how many leaves * A Full binary tree of height h will have how many leaves? * A Full binary tree of height h will have how many nodes * A Full binary tree of height h will have how many nodes?

Binary Search Trees / Slide 38 Complete Binary Trees * A complete binary tree (of height h) satisfies the following conditions: n Level 0 to h-1 represent a full binary tree of height h-1 n One or more nodes in level h-1 may have 0, or 1 child nodes n If j,k are nodes in level h-1, then j has more child nodes than k if and only if j is to the left of k

Binary Search Trees / Slide 39 Complete Binary Trees - Example

Binary Search Trees / Slide 40 Complete Binary Trees (cont) * Given a set of N nodes, a complete binary tree of these nodes provides the maximum number of leaves with the minimal average path length (per node) * The complete binary tree containing n nodes must have at least one path from root to leaf of length  log n 

Binary Search Trees / Slide 41 Height-balanced Binary Tree * A height-balanced binary tree is a binary tree such that: n The left & right subtrees for any given node differ in height by no more than one Each complete binary tree is a height- balanced binary tree * Note: Each complete binary tree is a height- balanced binary tree

Binary Search Trees / Slide 42 Height-balanced Binary Tree - Example

Binary Search Trees / Slide 43 Advantages of Height-balanced Binary Trees * Height-balanced binary trees are “balanced” * Operations that run in time proportional to the height of the tree are O(log n), n the number of nodeslimited performance variance * Operations that run in time proportional to the height of the tree are O(log n), n the number of nodes with limited performance variance * Variance is a very important concern in real time applications, e.g. connecting calls in a telephone network