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 12.2, Problem 8E
Program Plan Intro
To show that k successive calls to TREE-SUCCESSOR takes
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Prove that the height of a complete binary tree with n nodes is exactly ceil(lg(n+1))-1 using mathematical induction.
An in-order tree walk of an n-node binary search tree can be implemented by finding the minimum elementin the tree with TREE-MINIMUM and then making n-1 calls to TREE-SUCCESSOR. Prove that this algorithm runs inΘ(n) time.
(a) Show that a complete binary tree of height h has 2h+1 − 1 vertices. (b) Show that a nonempty binary tree with n vertices has height at least floor(log(n)).
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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- Prove that, if values are distinct, any binary search tree can be constructed by appropriately ordering insertion operations.arrow_forwardProve that a binary tree with k leaves has height at least log karrow_forwardshow that one can always build a red-black tree with n nodes such that there's no more than O(log n) red nodesarrow_forward
- Prove that efficient computation of the height of a BinaryTree musttake time proportional to the number of nodes in the tree.arrow_forwardProve that efficient computation of the height of a BinaryTree must take time proportional to the number of nodes in the treearrow_forwardUse the procedure TREE-SUCCESSOR and TREE-MINIMUM to write a function of x, x is a node in a binary search tree, to produce the output that INORDERTREE-WALK function would produce. Determine the upper bound running time complexity of F(x) and prove its correctness.arrow_forward
- Given an empty Binary Search Tree (BST). After performing the following insertion operations, INSERT 78 23 11 44 22 45 98 2 16 what is the right child of 11 in the resulting BST?arrow_forwardGiven the following Binary Tree, what is the result of a post-order traversal? What is the height of node N?arrow_forwardLet T be an n-node binary tree that may be improper. Describe how to represent T by means of a proper binary tree T' with O(n) nodes. python code pleasearrow_forward
- Illustrate that via AVL single rotation, any binary search tree T1 can betransformed into another search tree T2 (with the same items) Give an algorithm to perform this transformation using O(N log N) rotation on averagearrow_forwardLet 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_forwardImplement straight forward recursive algorithm to check if a given binary tree is BST?arrow_forward
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