A nonempty simple graph is a tree if and only if each pair of vertices is joined by exactly one path

Given: A nonempty simple graph.

A cycle is a closed walk of length at least three with distinct vertices apart from its endpoints; a forest is a simple graph with no cycle and a tree is a connected forest (Cycles, trees and forests in a simple graph on an arbitrary vertex set).

A walk of length $\mathrm{\ell }$ in a simple graph is a finite vertex list $(v_0,\dots ,v_{\mathrm{\ell }})$ with consecutive vertices adjacent; a path is a walk with distinct vertices; the graph is connected when it is nonempty and every two vertices are joined by a path (Walks, paths, connectedness and components in a simple graph on an arbitrary vertex set).

A simple graph is a pair $(V,E)$ with $V$ any set and $E$ a set of two-element subsets of $V$ (Simple graphs on an arbitrary vertex set).

Every walk in a simple graph contains a path with the same endpoints and of no greater length (Every walk contains a path between the same endpoints, of no greater length).