  Let $T$ be a tree with a vertex set $\{ 1,2,\dots, N \}$. Denote by $d_{ij}$
the distance between vertices $i$ and $j$. In this paper, we present an
explicit combinatorial formula of principal minors of the matrix
$(t^{d_{ij}})$, and its applications to tropical geometry, study of
multivariate stable polynomials, and representation of valuated matroids. We
also give an analogous formula for a skew-symmetric matrix associated with $T$.
