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Let T be a weighted tree with n leaves. Let D_{i,j} be the distance betweenthe leaves i and j. Let D_{i,j,k}= (D_{i,j} + D_{j,k} +D_{i,k})/2. We will callsuch numbers "triple weights" of the tree.  In this paper we find necessary and sufficient conditions for a set ofpositive real numbers indicized by 3-subsets of an n-set to be the set of thetriple weights of a tree with n leaves. By using the same ideas, we give also acharacterization, different from the previous ones, for sets indexed by2-subsets of a n-set to be double weights of a tree.

Sets of Double and Triple Weights of Trees