Question: Let $T$ be a tree, prove that at most $1$ perfect matching exists in $T$
My Proof:
Let $M$ and $M'$ be perfect matches in the tree $T = (V,E)$
And let $G$ be a graph on the vertex set $V$ and edges set: $M \cup M'$. Because $M$ and $M'$ cover all the vertices, each component of that graph $G$ is an isolated vertex (which is in $M$ and $M'$) or it is a cycle. Because $T$ is a tree, it does not have cycles and thus we conclude that $M = M'$
I would appreciate if you could tell me if I am wrong/right and tell me where should I fix that proof. Thank you!