Let $G$ a tree, $M$ maximum matching of $G$, then some leaf (vertex of degree 1) of $G$ is $M-$saturated.

Question

Let $G$ a tree, $M$ maximum matching of $G$, then some leaf (vertex of degree 1) of $G$ is $M-$saturated.

I am trying to do this problem by contradiction. Thus, since there are no M-saturated leaves, then if we take some vertex $x$ that is leaf, since it has degree 1, then there exists a vertex $y$ such that $xy\notin M$.

But I don't know if that result is useful to get the contradiction

Answer 1

Hint: if no leaf is saturated, pick two leaves and consider the unique path between them. Can you see how to create a larger matching?