/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 In each of \(35-50\) either draw... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 36

In each of \(35-50\) either draw a graph with the given specifications or explain why no such graph exists. Full binary tree, five intemal vertices, seven terminal vertices

Short Answer

Expert verified
In a full binary tree, there is a relationship between internal vertices (I) and terminal vertices (T) given by \(T = I + 1\). For the given specifications of 5 internal vertices and 7 terminal vertices, this relationship does not hold (7 鈮� 5 + 1). Therefore, no such graph of a full binary tree exists with the given specifications.

Step by step solution

01

Understand the relationship between internal and terminal vertices

In a full binary tree, there is a specific relationship between the number of internal vertices (I) and terminal vertices (T). This relationship can be expressed as follows: \[ T = I + 1 \] Applying this relationship to the given problem, we have: \[ 7 = 5 + 1 \] Step 2: Assessing if the relationship holds true
02

Check if the relationship is true

Notice that the relationship for a full binary tree does not hold true for the given exercise (7 does not equal to 5 + 1). Therefore, it is impossible to construct a full binary tree with 5 internal vertices and 7 terminal vertices. As a result, we can conclude that no such graph exists for the given specifications.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Binary Tree
A binary tree is a foundational concept in graph theory. It is a tree data structure in which each node has at most two children, referred to as the left and right child. In a binary tree structure:
  • The topmost node is called the root.
  • Nodes that have no children are called leaf nodes or terminal vertices.
  • Nodes that have at least one child are referred to as internal vertices.
Binary trees are widely used in computer science for their efficiency in storing data that can be referenced, inserted, or deleted quickly. Understanding how these trees are structured is crucial for designing algorithms that solve complex problems effectively. By only having two possible child nodes, binary trees are simpler and often more manageable compared to trees with many branches.
Internal Vertices
Internal vertices of a binary tree are nodes that have at least one child. They essentially steer the structure of the tree, determining the way data is organized and accessed.
  • An internal vertex is not a leaf or terminal vertex, meaning it is not at the end of a branching path.
  • In the context of a full binary tree, which is a special type of binary tree, each internal vertex has exactly two children.
Internal vertices play a crucial role in binary trees as they connect leaf nodes and ensure that every flow of data or path has a continuous chain of command from the root to the leaves. Recognizing them helps in assessing the depth and balance of the tree.
Full Binary Tree
A full binary tree is a unique type of binary tree where every node other than the leaves has two children. Full binary trees have very specific characteristics:
  • The total number of terminal vertices is always one more than the number of internal vertices.
  • This characteristic can be expressed mathematically as: \[T = I + 1\]
  • They are perfectly balanced, except for the terminal nodes, which ensures optimal performance in data operations.
In exploring the given problem, understanding this relationship is crucial because it lays the foundation to check if any given specifications fit the definition. The specificity of a full binary tree means any deviation will disrupt its balance and hence its recognition as a full binary tree.
Terminal Vertices
In a binary tree, terminal vertices, often called leaf nodes, are the end points and have no children. They form the bottom layer of the tree structure.
  • Terminal vertices are essential for completing the paths in a tree, delineating where operations or data retrieval terminate.
  • In a full binary tree, they always outnumber internal vertices by exactly one, which is a hallmark of such structures.
Understanding terminal vertices is vital because they dictate the possible configurations and depth of the tree. Every additional terminal vertex increases the tree's coverage area, impacting its efficiency and complexity. Therefore, in scenarios where the count doesn't match expected data, especially in full binary trees, it becomes immediately evident there's an anomaly in the tree's structure.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In each of \(35-50\) either draw a graph with the given specifications or explain why no such graph exists. Full binary tree, four internal vertices

In each of 8-21, either draw a graph with the given specifications or explain why no such graph exists. Simple graph, connected, six vertices, six edges

Each of (a)-(c) describes a graph. In each case answer yes, no, or not necessarily to this question: Does the graph have an Euler circuit? Justify your answers. a. \(G\) is a connected graph with five vertices of degrees 2 , \(2,3,3\), and 4 . b. \(G\) is a connected graph with five vertices of degrees 2, \(2,4,4\), and 6 . c. \(G\) is a graph with five vertices of degrees \(2,2,4,4\), and 6 .

Find graphs that have the following adjacency matrices. a. \(\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 0\end{array}\right] \quad\) b. \(\left[\begin{array}{lll}0 & 2 & 0 \\ 2 & 1 & 0 \\\ 0 & 0 & 1\end{array}\right]\)

Prove that if \(\mathbf{A}\) is an \(m \times m\) symmetric matrix, then \(\mathbf{A}^{2}\) is symmetric.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.