If a tree $T$ contains a perfect matching, then $n(T)$ is even.

Question

I would like to use this lemma in a proof, though I would like to vet it out first.

Proof: Assuming that $T$ contains a perfect matching, $M$, we have that $M$ saturates every vertex in $T$. Since there are two vertices for each edge, $n(T)=2|M|$. Furthermore, any integer multiplied by two is even, thus we have that $n(T)$ is even.

Thoughts?