Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 12.4, Problem 2E
Program Plan Intro
To give an asymptotic upper bound on the height of binary search tree with n node where average height of a node is
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Consider a binary search tree (BST) whose elements are the integer values:
Starting with an empty BST, show the effect of successively adding the following elements: 15, 78, 10, 5, 12, 82, 60, 68, 11 and 65.
Show the effect of successively deleting 10, 78, and 15 from the resulting BST.
Traverse the original BST in pre-order and post-order.
Implement a function to verify if a binary tree is balanced. A balanced tree, for the purposes of this question, is one in which the heights of the two subtrees of any node never differ by more than one.
Suppose T is a binary tree with 17 nodes. What is the minimum possible depth of T?
1
3
4
2
A binary tree of N nodes has _______.
Log2 N levels
N / 2 levels
Log10 N levels
N x 2 levels
The difference between a binary tree and a binary search tree is that :
in binary search tree nodes are inserted based on the values they contain
none of these
in binary tree nodes are inserted based on the values they contain
a binary search tree has two children per node whereas a binary tree can have none, one, or two children per node
What is the best code for the following procedure:
AddStudent(studentName):add a new student to an array of alphabetically ordered names . Hint: We must shift some students. size contains the number of students in the array
AddStudent(studentName){
int i ;
for( i=0; i< size-1; i++){
if(arr[i].compareTo( studentName)>0)
break;
for(int j= size-1 ; j >i…
Chapter 12 Solutions
Introduction to Algorithms
Ch. 12.1 - Prob. 1ECh. 12.1 - Prob. 2ECh. 12.1 - Prob. 3ECh. 12.1 - Prob. 4ECh. 12.1 - Prob. 5ECh. 12.2 - Prob. 1ECh. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - Prob. 5E
Ch. 12.2 - Prob. 6ECh. 12.2 - Prob. 7ECh. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.3 - Prob. 1ECh. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.4 - Prob. 1ECh. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12 - Prob. 1PCh. 12 - Prob. 2PCh. 12 - Prob. 3PCh. 12 - Prob. 4P
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- Draw the binary search tree that would result if the given elements were added to an empty binary search tree in the given order. Use paint to draw. a. Lisa, Bart, marge, maggie, flanders, smithers, miljouse b. 12,34,1,5,-5,6,19,45,2,-7,47arrow_forwardGive an argument for why the Prim's algorithm will always return a Minimum Spanning Tree?arrow_forwardConsider a weight balanced tree such that, the number of nodes in the left sub tree is at least half and at most twice the number of nodes in the right sub tree. The maximum possible height (number of nodes on the path from the root to the farthest leaf) of such a tree on k nodes can be described asa) log2 nb) log4/3 nc) log3 nd) log3/2 narrow_forward
- Let T be an arbitrary splay tree storing n elements A1, A2, . An, where A1 ≤ A2 ≤ . . . ≤ An. We perform n search operations in T, and the ith search operation looks for element Ai. That is, we search for items A1, A2, . . . , An one by one. What will T look like after all these n operations are performed? For example, what will the shape of the tree be like? Which node stores A1, which node stores A2, etc.? Prove the answer you gave for formally. Your proof should work no matter what the shape of T was like before these operations.arrow_forward1. Find the Big-O time complexity for the following:-Inserting an element in an (unbalanced) binary search tree? And what is the best case?-Best case for the height of an (unbalanced) binary search tree? And what is the average case big-O given the avg height if elements are inserted in random order?2. In a binary search tree of N nodes, how many subtrees are there?arrow_forwardDraw the portion of the state space tree generated by LCBB for the following instances. n = 4, m = 15, (P₁, ..., P) = (10, 10, 12, 18) (w₁,..... W 4) = (2, 4, 6, 9).arrow_forward
- Your colleague proposed a different definition of a binary search tree: it is such binary tree with keys in the nodes that for each node the key of its left child (if exists) is bigger than its key, and the key of its right child (if exists) is less than its key. Is this a good definition for a binary search tree? A. Yes B. Noarrow_forwardRemember, in our definition, the height of a binary tree means maximum number of nodes from the root to a leaf. a) In a perfect binary tree of size n, what is the tree's exact height? Justify your answer. b) In a degenerate binary tree of size n, what is the tree's exact height? Justify your answer. c) What is the maximum height for a balanced binary tree of size 7? Justify your answerarrow_forwardConsider the array t = [1, 2, 3, 4, 5, 8, 0 , 7, 6] of size n = 9, . a) Draw the complete tree representation for t. b) What is the index of the first leaf of the tree in Part a (in level order)? In general, give a formula for the index of the first leaf in the corresponding complete binary tree for an arbitrary array of size n. c) Redraw the tree from Part a after each call to fixheap, in Phase 1 of heapsort. Remember, the final tree obtained will be a maxheap. d) Now, starting with the final tree obtained in Part c, redraw the tree after each call to fixheap in Phase 2 of heap sort. For each tree, only include the elements from index 0 to index right (since the other elements are no longer considered part of the tree). e) For the given array t, how many calls to fixheap were made in Phase 1? How many calls to fixheap were made in Phase 2? f) In general , give a formula for the total number of calls to fixheap in Phase 1, when heapsort is given an arbitrary array of size n. Justify…arrow_forward
- Prove that the depth of a random binary search tree (depth of the deepest node) is O(logN), on average.arrow_forwardConsider the Binary Search Tree method DepthEqual(T) that outputs all node values in a binary search tree T such that the value stored in the node is equal to the depth of the node in the tree. The values can be any integer, including negative numbers. Show pseudocode for an efficient implementation of DepthEqual(T). Hint: For a node n at depth d, whose value is v. if d >= v. then there is no need to search n's left subtreearrow_forwardCan you explain the difference between a spanning tree and an MST? Is there a similar comparison and analysis of Prim and Kruskal's algorithms?arrow_forward
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