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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

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

Suppose that \({{\bf{d}}_{\bf{1}}}{\bf{,}}{{\bf{d}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{d}}_{\bf{n}}}\) are n positive integers with sum \({\bf{2n - 2}}\). Show that there is a tree that has n vertices such that the degrees of these vertices are \({{\bf{d}}_{\bf{1}}}{\bf{,}}{{\bf{d}}_{\bf{2}}}{\bf{,}}...{\bf{,}}{{\bf{d}}_{\bf{n}}}\).

Draw the subtree of the tree in Exercise \(4\)that is rooted at

\(\begin{array}{l}{\bf{a}})a.\\b)c.\\c)e.\end{array}\)

a) What is the height of a rooted tree?

b) What is a balanced tree?

c) How many leaves can an \({\bf{m}}\)-ary tree of height \({\bf{h}}\) have?

Prove that the reverse-delete algorithm always producesa minimum spanning tree when given as input a weightedgraph with distinct edge weights. (Hint: Use Exercise \(33\).)

a) Define a rooted tree and the root of such a tree.

b) Define the parent of a vertex and a child of a vertex in a rooted tree.

c) What are an internal vertex, a leaf, and a subtree in a rooted tree\(?\)

d) Draw a rooted tree with at least \({\bf{10}}\) vertices, where the degree of each vertex does not exceed \({\bf{3}}\). Identify the root, the parent of each vertex, the children of each vertex, the internal vertices, and the leaves.
See all solutions

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