Question 116
$S$ and $T$ are two trees without having a vertex of degree $2$. Each edge of them has a positive number named the *length* of the edge. The *distance* between two vertices is the sum of the *length* of the edges of the path between them. We call the vertices with degree $1$, *leaf*. $f$ is an injective and surjective function from the set of *leaves* of $S$ to the set of *leaves* of $T$ with this property that for every two *leaves* $u$ and $v$ in $S$, the *distance* between $u$ and $v$ in $S$ is equal to the *distance* between $f(u)$ and $f(v)$ in $T$. Prove that there exists an injective and surjective function $g$ from the set of vertices of $S$ to the set of vertices of $T$ such that for every two vertices $u$ and $v$ in $S$, the *distance* between $u$ and $v$ in $S$, is equal to the *distance* between $g(u)$ and $g(v)$ in $T$.