Graham-Pollak determinant formula

Theorem (Graham, Pollak; 1971): Let $D_n$ be the distance matrix of a tree, i.e.,

\[(D_n)_{xy} = d(x,y).\]

Then

\[\det(D_n) = (-1)^{n-1}(n-1)2^{n-2}.\]

Thus, surprisingly, the the determinant of $D_n$ dopends solely on on the number of the vertices of a tree and not at all on its structure.

Is there some good combinatorial explanation of this formula?