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Chapter 12: Problem 12

a) Let \(T=(V, E)\) be a binary tree. If \(|V|=n\), what is the maximum height that \(T\) can attain? b) If \(T=(V, E)\) is a complete binary tree and \(|V|=n\), what is the maximum height that \(T\) can reach in this case?

Short Answer

Expert verified
a) The maximum height that a binary tree with n vertices can attain is \(n-1\). b) The maximum height that a complete binary tree with n vertices can reach is approximately \( \lfloor log_2(n+1) \rfloor \), the integer part of \(log_2(n+1)\).

Step by step solution

01

Maximum height of a binary tree

For a binary tree \(T=(V, E)\) with \(|V|=n\) (n vertices), the maximum height is attained when the binary tree degenerates into a linear chain of n nodes. This means that each node (except for the last one) is only connected to one child. The height of a tree is the number of edges in the longest path from the root to a leaf. Hence, the maximum height of a binary tree would be \(n-1\) when it forms a linear chain of nodes.
02

Maximum height of a complete binary tree

In the case of a complete binary tree, all levels of the tree are fully filled except possibly for the last level, which is filled from left to right. In a complete binary tree, the number of vertices n is roughly \(2^h+1-1\) where h is the height of the tree. To find the maximum height, we solve for \(h\) in terms of \(n\): \(h = \lfloor log_2(n+1) \rfloor \). Thus, in a complete binary tree, the maximum height is approximately \( \lfloor log_2(n+1) \rfloor \) which is the integer part of the logarithm base 2 of \(n+1\).

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Most popular questions from this chapter

Let \(L_{i}\), for \(1 \leq i \leq 4\), be four lists of numbers, each sorted in ascending order. The numbers of entries in these lists are \(75,40,110\), and 50 , respectively. a) How many comparisons are needed to merge these four lists by merging \(L_{1}\) and \(L_{2}\), merging \(L_{3}\) and \(L_{4}\), and then merging the two resulting lists? b) How many comparisons are needed if we first merge \(L_{1}\) and \(L_{2}\), then merge the result with \(L_{3}\), and finally merge this result with \(L_{4}\) ? c) In order to minimize the total number of comparisons in this merging of the four lists, what order should the merging follow? d) Extend the result in part (c) to \(n\) sorted lists \(L_{1}, L_{2}, \ldots, L_{n}\).

Let \(T=(V, E)\) be a tree with \(|V|=n \geq 2\). How many distinct paths are there (as subgraphs) in \(T\) ?

If \(B_{1}, B_{2}, \ldots, B_{k}\) are the biconnected components of a loop-free connected undirected graph \(G\), how is \(\chi(G)\) related to \(\chi\left(B_{i}\right), 1 \leq i \leq k\) ?

a) Let \(T=(V, E)\) be a tree. If \(v \in V\), prove that \(v\) is an articulation point of \(T\) if and only if \(\operatorname{deg}(v)>1\). b) Let \(G=(V, E)\) be a loop-free connected undirected graph with \(|E| \geq 1\). Prove that \(G\) has at least two vertices that are not articulation points.

a) If \(G=(V, E)\) is a forest with \(|V|=v,|E|=e\), and \(\kappa\) components (trees), what relationship exists among \(v, e\), and \(\kappa\) ? b) What is the smallest number of edges we must add to \(G\) in order to get a tree?
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