Graph Theory - Edge-Disjoint Paths

Question

Let $G$ be a forest with exactly $2k$ vertices of odd degree. Prove that there exist $k$ edge-disjoint paths whose union is $G$.

I am not sure what to do, could someone please help me?

Answer 1

Hint: Let $W$ be a path of maximal length in $G$. What can you say about $G-W$?

Answer 2

So according to the hint given above, we remove the edges in the maximal length path $P$ of $G$. By doing this, the beginning and the end vertices of the path will become even degree vertex. Given that $G$ is a tree and the path is maximal, the beginning and the end vertices of the path are leaves in the graph. And so discounting vertices without edges connected to them, $G-P$ is a tree with $2k-2$ vertices of odd degree. We can repeate the process to get $k$ disjoint path.