I have to show that a tree $T = (V,E)$ has at most one perfect matching.
In order to show that, one has to assume that the tree $T$ has an even number of vertices, so that a perfect matching can exist. Due to that I think that one should work inductively by adding an even number of vertices (at least two) to the existing tree and showing that they "become" part of the already existing matching. The thing is the formality..
Thank you for your help
Best Answer
You are on the right track. Let $v$ be a leaf vertex with parent $u$. You should show that $T$ has a perfect matching if and only if $T - \{u,v\}$ has a perfect matching, and more precisely, the perfect matchings of these graphs are in a one-to-one correspondence. This will let you use induction on the number of vertices. Note that $T - \{u,v\}$ may be disconnected, but this is not a problem if you do the induction for forests instead of trees.