A characterization of dissimilarity families of trees

In this paper we find some equalities and inequalities characterizing the families of real numbers parametrized by the k-subsets of {1,…,n} that are the families of k-weights of weighted trees whose leaf set is equal to {1,…,n} and whose weights of the internal edges are positive (where we say that an edge e is internal if there exists a path with endpoints of degree greater than 2 and containing e).