A forest is a graph that contains no cycles, and a connected forest is a tree. For example, Fig. 1.1 shows a forest with four components, each of which is a tree

". Note that trees and forests are simple graphs.

                                                               Fig. 1.1

Another definition

A forest is an acyclic graph. The connected components of a forest are therefore trees, which explains the use (very natural!) of the terminology.

Forests generalize trees.

Proposition 1.1. In a forest G, we have m ≤ n − 1, with equality if and only if G is a tree.

Proof. Given C₁, C₂, ..., C_p the connected components of G, apply proposition( If G is a tree then m = n − 1.) to each of these components, denoting respectively n_i and

m_i the number of vertices and the number of edges of C_i. By adding all these equalities, for i =1, 2,...,p, we then have:

Thus:

                                m = n − p

and since p ≥ 1(p is the number of connected components of G):

                               m ≤ n − 1

Equality occurs if and only if p = 1, that is if and only if G is connected. Since G is by hypothesis acyclic, this means if and only if G is a tree.

Graph Theory  and Applications ,Jean-Claude Fournier, WILEY, page(47)

Introduction to  Graph Theory  ,Fourth edition,  Robin J. Wilson, 1998