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Chapter 11: Problem 25

Either draw a full \(m\) -ary tree with 84 leaves and height 3 , where \(m\) is a positive integer, or show that no such tree exists.

Short Answer

Expert verified
No such tree exists since the cube root of 84 is not an integer.

Step by step solution

01

Understand the Tree Structure

An m-ary tree is a tree where each node has at most m children. We are given that the height of the tree is 3 and it has 84 leaves.
02

Determine the Number of Nodes per Level

A full m-ary tree of height 3 will have nodes at levels 0, 1, 2, and 3. At level 0 (the root), there is 1 node. At level 1, there are m nodes. At level 2, there are m^2 nodes, and at level 3, there are m^3 leaves.
03

Relate Leaves to Nodes

We are told that the number of leaves (nodes at height 3) is 84. Therefore, the equation to solve is: \[ m^3 = 84 \]
04

Solve for m

To find m, we solve the equation: \[ m^3 = 84 \]\[ m = \root{3}{84} \]The cube root of 84 is approximately 4.38, which is not an integer. Therefore, no such integer m exists.
05

Conclusion

Since 4.38 is not a positive integer, it's not possible to have a full m-ary tree with exactly 84 leaves and height 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tree Structure
A tree structure is a foundational concept in computer science and mathematics. It simply consists of nodes connected by edges, resembling a tree from nature, with a root at the top and branches extending downward.
In an m-ary tree, each node can have up to m children. This means every node in the tree can branch out to m other nodes. Trees are used extensively to model hierarchical data, such as file systems, organizational structures, or expressions in programming.
The 'full' m-ary tree term means every node has either 0 or m children. This ensures a uniform structure, simplifying analysis and algorithms involving the tree.
Height of Tree
The height of a tree is the length of the longest path from the root to any leaf node. To be more specific, the height is the number of edges in this path.
In the given problem, the tree has a height of 3. This refers to a structure with node levels as follows:
- The root at level 0
- The first layer of children at level 1
- The second layer of children at level 2
- The final layer at level 3, which comprises the tree's leaves
Understanding tree height is crucial because it affects both the performance of tree operations and the overall structure.
Number of Nodes
The number of nodes in a tree indicates both the size and complexity of the tree. Nodes can either be internal (having children) or external (leaves).
To determine the nodes per level in an m-ary tree, follow these general rules:
  • The root (level 0) has 1 node.
  • The first level has m nodes.
  • The second level has m虏 nodes.
  • The third level (given in the problem) has m鲁 nodes.
This is calculated exponentially concerning the level's depth. For the problem at hand, it was observed that the third level (height 3) has 84 nodes, and setting up the equation as \( m^3 = 84 \), it was found that m must be around 4.38. Since m should be a positive integer, no such m-ary tree with exactly 84 leaves and height 3 can exist.
Understanding the number of nodes in each level helps in visualizing and validating tree structures in problems.

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Most popular questions from this chapter

Devise an algorithm similar to Prim鈥檚 algorithm for constructing a maximum spanning tree of a connected weighted graph.

Let \(G\) be a simple graph with \(n\) vertices. Show that a) \(G\) is a tree if and only if it is connected and has \(n-1\) edges. b) \(G\) is a tree if and only if \(G\) has no simple circuits and has \(n-1\) edges. \([\text { Hint: To show that } G \text { is connected }\) if it has no simple circuits and \(n-1\) edges, show that \(G\) cannot have more than one connected component. \(]\)

Can the leaves of an ordered rooted tree have the following list of universal addresses? If so, construct such an ordered rooted tree. a) \(1.1 .1,1.1 .2,1.2,2.1 .1 .1,2.1 .2,2.1 .3,2.2,3.1 .1\) \(3.1 .2 .1,3.1 .2 .2,3.2\) b) \(1.1,1.2 .1,1.2 .2,1.2 .3,2.1,2.2 .1,2.3 .1,2.3 .2\) 2.4.2.1, \(2.4 .2 .2,3.1,3.2 .1,3.2 .2\) c) \(1.1,1.2 .1,1.2 .2,1.2 .2 .1,1.3,1.4,2,3.1,3.2,4.1 .1 .1\)

a) How many nonisomorphic unrooted trees are there with four vertices? b) How many nonisomorphic rooted trees are there with four vertices (using isomorphism for directed graphs)?

The rooted Fibonacci trees \(T_{n}\) are defined recursively in the following way. \(T_{1}\) and \(T_{2}\) are both the rooted tree consisting of a single vertex, and for \(n=3,4, \ldots,\) the rooted tree \(T_{n}\) is constructed from a root with \(T_{n-1}\) as its left subtree and \(T_{n-2}\) as its right subtree. How many vertices, leaves, and internal vertices does the rooted Fibonacci tree \(T_{n}\) have, where \(n\) is a positive integer? What is its height?
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