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Chapter 11: Q1RE (page 804)

a. Define a tree.

b. Define a forest.

Short Answer

Expert verified

a. A tree is an undirected graph that is connected and that does not contain any simple circuits.

b. A forest is a group of trees that are not connected with each other.

Step by step solution

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01

Path

A simple path is a path that does not contain the same edge more than once.

A circuit is a path that begins and ends in the same vertex.

A graph is connected if there exists a path between every pair of vertices.

A tree is an undirected graph that is connected and that does not contain any simple circuits.

For example:

02

Forest

A forest is a group of trees that are not connected with each other.

For example:

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Most popular questions from this chapter

Give a definition of well-formed formulae in postfix notation over a set of symbols and a set of binary operators.

Extend the definition of well-formed formulae in prefix notation to sets of symbols and operators where the operators may not be binary.

a. Explain how backtracking can be used to determine whether a simple graph can be colored using \(n\) colors.

b. Show, with an example, how backtracking can be used to show that a graph with a chromatic number equal to \({\bf{4}}\) cannot be colored with three colors, but can be colored with four colors.

Show that any well-formed formula in prefix notation over a set of symbols and a set of binary operators contains exactly one more symbol than the number of operators.

What is wrong with the following “proof” using mathematical induction of the statement that every tree with \({\bf{n}}\) vertices has a path of length \({\bf{n - 1}}\). Basis step: Every tree with one vertex clearly has a path of length \(0\). Inductive step: Assume that a tree with \({\bf{n}}\) vertices has a path of length \({\bf{n - 1}}\), which has \({\bf{u}}\) as its terminal vertex. Add a vertex \({\bf{v}}\) and the edge from \({\bf{u}}\)to \({\bf{v}}\). The resulting tree has \({\bf{n + 1}}\) vertices and has a path of length \({\bf{n}}\). This completes the inductive step.
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