[Math] Perfect matching in a tree

Question

Prove that in a tree there is at most $1$ perfect matching

If the number of vertices is even$\implies$ number of edges odd, not divisible by $2$, so no perfect matching

For the other case can you apply induction using $2$ leaves ?

Answer 1

Applying induction by removing a leaf is the right idea. If $x$ is a leaf, and the edge meeting $x$ is $xy$, then any perfect matching for $T$ must consist of $xy$ together with a perfect matching of $T-\{x,y\}$. Now $T-\{x,y\}$ isn't necessarily a tree, but all of its components are trees.