IR-TST 2010 Problem 4

$S,T$ are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices $v,w$ in this graph is defined by sum of length of all edges in the path between $v$ and $w$ . Let $f$ be a bijective function from leaves of $S$ to leaves of $T$ , such that for each two leaves $u,v$ of $S$ , distance of $u,v$ in $S$ is equal to distance of $f(u), f(v)$ in $T$ . Prove that there is a bijective function $g$ from vertices of $S$ to vertices of $T$ such that for each two vertices $u,v$ of $S$ , distance of $u,v$ in $S$ is equal to distance of $g(u)$ and $g(v)$ in $T$ .