A characterization of dissimilarity families of trees

Let {cal T}=(T,w) be a weighted finite tree with leaves 1,..., n. For any I :={i_1,..., i_k } subset of {1,...,n}, let D_I ({cal T}) be the weight of the minimal subtree of T connecting i_1,..., i_k; the D_{I} ({cal T}) are called k-weights of {cal T}. Let {D_I}_{I k-subset of {1,...,n}} be a family of real numbers. We say that a weighted tree {cal T}=(T,w) with leaves 1,..., n realizes the family if D_I({cal T})=D_I for any k-subset I of {1,...,n}. 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).