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Chapter 12: Problem 19

What is the maximum number of internal vertices that a complete quaternary tree of height 8 can have? What is the number for a complete \(m\)-ary tree of height \(h\) ?

Short Answer

Expert verified
The maximum number of internal vertices a complete quaternary tree of height 8 can have is 87381 vertices. For a complete m-ary tree of height h, the maximum number of internal vertices it can have is given by the formula \(\frac{m^h - 1}{m - 1}\) vertices.

Step by step solution

01

Understand The Definition of a quaternary Tree

A quaternary (or 4-ary) tree is a tree in which each internal vertex has exactly four children. Similarly, an m-ary tree is a tree in which each internal vertex has exactly m children.
02

Find The Maximum Number of Internal Vertices of Quaternary Tree of Height 8

The number of internal vertices in a complete quaternary tree is given by the sum of vertices from level 0 to level h-1 (because the height is counted from 0). Therefore, the sum would be \(4^0 + 4^1 + 4^2 + . . . + 4^7 = 87381\), where each term represents the maximum number of nodes at each level.
03

Generalize for a Complete m-ary Tree of Height h

Similar to the above step, the number of internal vertices in a complete m-ary tree is given by the sum of vertices from level 0 to level h-1. Therefore, the sum would be \(m^0 + m^1 + . . . + m^{h-1}\), which is a geometric series formula. Sum of the first h terms of a geometric series is given by \(\frac{m^h - 1}{m - 1}\) if m>1, subsituting m and h, we get \(\frac{m^h - 1}{m - 1}\) number of internal nodes in a complete m-ary tree of height h.

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Most popular questions from this chapter

If \(G=(V, E)\) is a loop-free undirected graph, prove that \(G\) is a tree if there is a unique path between any two vertices of \(G\).

The connected undirected graph \(G=(V, E)\) has 30 edges. What is the maximum value that \(|V|\) can have?

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?

a) Let \(G=(V, E)\) be a loop-free undirected graph with \(|V|=n\). Prove that \(G\) is a tree if and only if \(P(G, \lambda)=\lambda(\lambda-1)^{n-1}\). b) Prove that \(\chi(G)=2\) for any tree with two or more vertices. c) If \(G=(V, E)\) is a connected undirected graph with \(|V|=n\), prove that for any integer \(\lambda \geq 0\), \(P(G, \lambda) \leq \lambda(\lambda-1)^{n-1}\).

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?
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