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1 A binary tree is 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 2h+1- 1 nodes and at most 2 leaves, and that it achieves these maxima when it is complete, meaning that it is full and all leaves are at the same distance from the root. Find v(h), the 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. Hint 1: try a few simple cases (h = 0,1,2, 3,...) and see if you can guess what v(h) is.

1 A binary tree is 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 2h+1- 1 nodes and at most 2 leaves, and that it achieves these maxima when it is complete, meaning that it is full and all leaves are at the same distance from the root. Find v(h), the 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. Hint 1: try a few simple cases (h = 0,1,2, 3,...) and see if you can guess what v(h) is.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A binary tree is 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 2h+1 - 1 nodes and at most 2* leaves, and that it achieves these maxima when it is complete, meaning that it is full 1 and all leaves are at the same distance from the root. Find v(h), the minimum number of leaves that a full binary tree of heighth can have, and prove your answer using ordinary induction on h. Note that tree of height of 0 is a single (leaf) node. Hint 1: try a few simple cases (h = 0,1,2, 3,...) and see if you can guess what v(h) is.
Transcribed Image Text:A binary tree is 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 2h+1 - 1 nodes and at most 2* leaves, and that it achieves these maxima when it is complete, meaning that it is full 1 and all leaves are at the same distance from the root. Find v(h), the minimum number of leaves that a full binary tree of heighth can have, and prove your answer using ordinary induction on h. Note that tree of height of 0 is a single (leaf) node. Hint 1: try a few simple cases (h = 0,1,2, 3,...) and see if you can guess what v(h) is.
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