Finite Connected Graph is Tree iff Size is One Less than Order/Lemma

Theorem
Let $G = \struct {V, E}$ be a non-edgeless connected finite simple graph.

Let $G$ have no cycles.

Then:
 * There exists at least one vertex of $G$ which is adjacent to exactly one other vertex of $G$.

Proof

 * : $\S 2.3.4.1$: Free Trees: Theorem $\mathrm A$