Families of multiweights and pseudostars

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 T   connecting i1,…,iki1,…,ik; the DI(T)DI(T) are called k  -weights of TT. Given a family of real numbers parametrized by the k  -subsets of {1,…,n}{1,…,n}, {DI}I∈({1,…,n}k), 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 I.In [14] Pachter and Speyer proved that, if 3≤k≤(n+1)/23≤k≤(n+1)/2 and {DI}I∈({1,…,n}k) is a family of positive real numbers, then there exists at most one positive-weighted essential tree TT with leaves 1,…,n1,…,n that realizes the family (where “essential” means that there are no vertices of degree 2). We say that a tree P   is a pseudostar of kind (n,k)(n,k) if the cardinality of the leaf set is n and any edge of P divides the leaf set into two sets such that at least one of them has cardinality ≥k  . Here we show that, if 3≤k≤n−13≤k≤n−1 and {DI}I∈({1,…,n}k) is a family of real numbers realized by some weighted tree, then there is exactly one weighted essential pseudostar P=(P,w)P=(P,w) of kind (n,k)(n,k) with leaves 1,…,n1,…,n and without internal edges of weight 0, that realizes the family; moreover we describe how any other weighted tree realizing the family can be obtained from PP. Finally we examine the range of the total weight of the weighted trees realizing a fixed family.