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 16.3, Problem 2E
Program Plan Intro
To prove that a binary tree that is not full cannot correspond to an optimal prefix code.
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Prove Proposition : In a 2-3 tree with N keys, search and insert operations are guaranteed to visit at least lg N nodes.
Prove that any binary tree of height h (where the empty tree is height 0, and a tree witha single node is height 1) has between h and 2h − 1 nodes, inclusive. A binary tree is onein which every node has at most three edges (at most one to the ’parent’ and two to the’children.’)
Prove that efficient computation of the height of a BinaryTree musttake time proportional to the number of nodes in the tree.
Chapter 16 Solutions
Introduction to Algorithms
Ch. 16.1 - Prob. 1ECh. 16.1 - Prob. 2ECh. 16.1 - Prob. 3ECh. 16.1 - Prob. 4ECh. 16.1 - Prob. 5ECh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5E
Ch. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16 - Prob. 1PCh. 16 - Prob. 2PCh. 16 - Prob. 3PCh. 16 - Prob. 4PCh. 16 - Prob. 5P
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- Prove that a binary tree with k leaves has height at least log karrow_forwardYour 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_forwardProve Proposition : Search and insert operations in a 2-3 tree with N keys are guaranteed to visit at most lg N nodes.arrow_forward
- What is the optimal binary tree with weights 15,20,40,50arrow_forwardA binary tree is \emph{full} if every non-leaf node has exactly two children. For context, recall that we saw in lecture that a binary tree of height $h$ can have at most $2^{h+1}-1$ nodes and at most $2^h$ leaves, and that it achieves these maxima when it is \emph{complete}, meaning that it is full and all leaves are at the same distance from the root. Find $\nu(h)$, the \emph{minimum} number of leaves that a full binary tree of height $h$ can have, and prove your answer using ordinary induction on $h$. Note that tree of height of 0 is a single (leaf) node. \textit{Hint 1: try a few simple cases ($h = 0, 1, 2, 3, \dots$) and see if you can guess what $\nu(h)$ is.}arrow_forwardIn a binary tree, each parent node can only have two descendants. Prove that in any binary tree, the number of nodes producing two offspring is exactly one less than the number of leaves.arrow_forward
- Given that a tree with a single node has a height of one, what is the maximum number of nodes that may be included in a balanced binary tree with a height of five?arrow_forwardProve that efficient computation of the height of a BinaryTree must take time proportional to the number of nodes in the treearrow_forwardDiscrete Mathematics Establish a prove that the inverse of Prufer algorithm produces a tree with the same Prufer code as the input.arrow_forward
- Given that a tree with a single node has a height of 1, how many nodes could possibly be included in a balanced binary tree with a height of 5?arrow_forwardA binary tree is a rooted tree in which each node may produce no more than two children. Prove that in any binary tree, the number of nodes that produce two offspring is exactly one less than the number of leaves.arrow_forwardGiven that a tree with a single node has a height of one, what is the maximum number of nodes that may be included in a balanced binary tree that has a height of 5?arrow_forward
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