On dissimilarity vectors of general weighted trees

Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set { 1 , … , n } . For any i 1 , … , i k ∈ { 1 , … , n } , define D i 1 , … , i k ( T ) to be the sum of the lengths of the edges of the minimal subtree joining i 1 , … , i k . We will call such numbers “ k -weights” of the tree and we call the k -weights for any k ≥ 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.