A tree is a connected undirected graph with no cycles.

This definition has several equivalent consequences. For a finite tree with $n$ vertices:

$$|E|=n-1.$$

There is also exactly one simple path between every pair of vertices.

Removing any edge disconnects a tree, while adding any new edge creates exactly one cycle. Trees are therefore minimally connected graphs.

A forest is an undirected graph with no cycles; each connected component of a forest is a tree.

Trees appear whenever a structure branches without reconverging: hierarchies, parse structures, file trees and spanning structures in networks all use the same mathematical object.