Given a Tree $T$ with max. degree $\Delta > 1$ show that there are at least $\Delta$ vertices with degree exactly one.
The solutions in the book I'm working with do not include (at least explicitly) my first idea/intuition wich is the following:
consider a vertex $u$ in $T$ with deg$(u)=\Delta$. Consider maximal paths in $T$ starting at $u$ and containing exactly one of the edges ending in $u$. There are $\Delta$ of them and each of them ends in a leaf. Also these paths do not cross otherwise they would from a cycle containing $u$ wich is not possible since $T$ is a Tree. So each of the $\Delta$ paths corresponds to exactly one leaf (of degree one), and we are done.
Question: is this a valid argument or am I missing something?