A characterization of dissimilarity families of trees

Abstract

Let T=(T,w) be a weighted finite tree with leaves 1,…,n. For any I≔{i1,…,ik}⊂{1,…,n}, let DI(T) be the weight of the minimal subtree of T connecting i1,…,ik; the DI(T) are called k-weights of T. Let {DI}Ik-subset of {1,…,n} be a family of real numbers. We say that a weighted tree T=(T,w) with leaves 1,…,n realizes the family if DI(T)=DI 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).