Principal minors of the distance matrix of a tree

Let $T = ([n], E)$ be a tree and let D = ( d(i,j) )i, j ≤ n be the distance matrix of T. Let S⊆ [n]. We give the first combinatorial proof for a formula to compute the principal minor of D indexed by S, namely D[S]. This generalizes work of Graham and Pollak, as well as more recent works.