Sets of double and triple weights of trees

Let T be a weighted tree with n leaves numbered by the set {1,...,n}. Let D_{i,j}(T) be the distance between the leaves i and j. Let D_{i,j,k}(T)= \frac{1}{2}( D_{i,j}(T) + D_{j,k}(T) +D_{i,k}(T)). We will call such numbers ``triple weights'' of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a n-set to be double weights of a tree. By using the same ideas, we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an n-set to be the set of the triple weights of a tree with n leaves. Besides we propose a slight modification of Saitou-Nei's Neighbour-Joining algorithm to reconstruct trees from the data D_{i,j}