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Chapter 11: Problem 17

How many edges does a tree with \(10,000\) vertices have?

Short Answer

Expert verified
A tree with 10,000 vertices has 9,999 edges.

Step by step solution

01

Understand the properties of a tree

A tree is an undirected graph in which any two vertices are connected by exactly one path. Additionally, a tree with n vertices always has n-1 edges.
02

Apply the properties

For a tree with n vertices, we can directly use the property that the number of edges is n-1.
03

Calculate the number of edges

Given that the tree has 10,000 vertices, the number of edges is: 10,000 - 1 = 9,999.

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Key Concepts

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

Tree Properties
A tree is a fundamental concept in graph theory. It is an undirected graph where any two vertices are connected by exactly one path. This means there are no cycles in a tree, making it a type of acyclic graph. A notable property of trees is that they are always connected; in other words, there is a path between any pair of vertices. Additionally, a tree with vertices will always have -1 edges. This is a key characteristic that helps in calculating edges and understanding tree structures.
A few important properties of trees include:
  • Uniqueness of Paths: Only one unique path exists between any two vertices.
  • Edge Count: Always has exactly -1 edges for vertices.
  • Acyclic Nature: Contains no cycles.
  • Connected Graph: All vertices are interconnected.
Recognizing these properties can help you easily solve many tree-related problems in graph theory.
Graph Theory
Graph theory is a branch of mathematics that studies the properties of graphs, which are structures made up of vertices (nodes) connected by edges (lines). Trees are a special type of graph in this field.
In graph theory, it's crucial to understand the different types of graphs and their characteristics. Some common types include:
  • Directed Graphs: Edges have a direction from one vertex to another.
  • Undirected Graphs: Edges do not have a direction, like in trees.
  • Complete Graphs: Every pair of distinct vertices is connected by a unique edge.
  • Connected Graphs: There is a path between any pair of vertices.
  • Acyclic Graphs: Graphs without cycles.

Trees fit into the category of undirected, connected, and acyclic graphs. Understanding these classifications helps in analyzing and solving problems involving trees in graph theory. Moreover, it aids in grasping their unique properties and applications.
Vertices and Edges in Trees
In the context of trees, vertices and edges have a straightforward but essential relationship. The number of edges in a tree is always one less than the number of vertices. So, if a tree has vertices, it will have -1 edges.
Consider a tree with 10,000 vertices. Using the property mentioned above:
- 1 = 10,000 - 1 = 9,999.
This means the tree will have 9,999 edges.
The relationship between vertices and edges is pivotal for solving many problems in graph theory related to trees. To summarize:
  • A tree with vertices always has -1 edges.
  • This relationship simplifies calculations and understanding of tree structures.
  • It ensures that trees are both connected and acyclic, which is foundational in their definition.
Understanding this relationship is crucial for analyzing trees and solving related mathematical problems.

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Most popular questions from this chapter

a) Use Huffman coding to encode these symbols with frequencies a: 0.4, b: 0.2, c: 0.2, d: 0.1, e: 0.1 in two different ways by breaking ties in the algorithm differently. First, among the trees of minimum weight select two trees with the largest number of vertices to combine at each stage of the algorithm. Second, among the trees of minimum weight select two trees with the smallest number of vertices at each stage. b) Compute the average number of bits required to encode a symbol with each code and compute the variances of this number of bits for each code. Which tie-breaking procedure produced the smaller variance in the number of bits required to encode a symbol?

In Exercises \(13-15\) use depth-first search to produce a spanning tree for the given simple graph. Choose \(a\) as the root of this spanning tree and assume that the vertices are ordered alphabetically.

How many weighings of a balance scale are needed to find a counterfeit coin among 12 coins if the counterfeit coin is lighter than the others? Describe an algorithm to find the lighter coin using this number of weighings.

Suppose that we vary the payoff to the winning player in the game of nim so that the payoff is n dollars when n is the number of legal moves made before a terminal position is reached. Find the payoff to the first player if the initial position consists of a) two piles with one and three stones, respectively. b) two piles with two and four stones, respectively. c) three piles with one, two, and three stones, respectively.

Show that an ordered rooted tree is uniquely determined when a list of vertices generated by a preorder traversal of the tree and the number of children of each vertex are specified.
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