Path in Tree is Unique/Sufficient Condition

Theorem
Let $T$ be a graph.

Let $T$ be such that between any two vertices there is exactly one path.

Then $T$ is a tree.

Proof
Let $T$ be such that between any two vertices there is exactly one path.

Then for a start $T$ is by definition connected.

Suppose $T$ had a circuit, say $\left({u, u_1, u_2, \ldots, u_n, v, u}\right)$.

Then there are two paths from $u$ to $v$:
 * $\left({u, u_1, u_2, \ldots, u_n, v}\right)$

and
 * $\left({u, v}\right)$.

Hence, by Modus Tollendo Tollens, $T$ can have no circuits.

That is, by definition, $T$ is a tree.