On dissimilarity vectors of general weighted trees

Let TT be a (not necessarily positive) weighted tree with nn leaves numbered by the set {1,…,n}{1,…,n}. For any i1,…,ik∈{1,…,n}i1,…,ik∈{1,…,n}, define Di1,…,ik(T)Di1,…,ik(T) to be the sum of the lengths of the edges of the minimal subtree joining i1,…,iki1,…,ik. We will call such numbers “kk-weights” of the tree and we call the kk-weights for any k≥2k≥2 “multi-weights” of the tree. In this paper, we give a characterization of the families of real numbers that are the families of the multi-weights of a tree.