Let T=(T,w)T=(T,w) be a weighted finite tree with leaves 1,…,n1,…,n. For any I≔{i1,…,ik}⊂{1,…,n}I≔{i1,…,ik}⊂{1,…,n}, let DI(T)DI(T) be the weight of the minimal subtree of TT connecting i1,…,iki1,…,ik; the DI(T)DI(T) are called kk-weights of TT. Let {DI}Ik-subset of  {1,…,n} be a family of real numbers. We say that a weighted tree T=(T,w)T=(T,w) with leaves 1,…,n1,…,n realizes the family if DI(T)=DIDI(T)=DI for any kk-subset II of {1,…,n}{1,…,n}.#R##N##R##N#In this paper we find some equalities and inequalities characterizing the families of real numbers parametrized by the kk-subsets of {1,…,n}{1,…,n} that are the families of kk-weights of weighted trees whose leaf set is equal to {1,…,n}{1,…,n} and whose weights of the internal edges are positive (where we say that an edge ee is internal if there exists a path with endpoints of degree greater than 2 and containing ee).
['Agnese Baldisserri', 'Elena Rubei']
A characterization of dissimilarity families of trees