Abstract
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 .
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Rubei, E. Sets of Double and Triple Weights of Trees. Ann. Comb. 15, 723–734 (2011). https://doi.org/10.1007/s00026-011-0118-1
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DOI: https://doi.org/10.1007/s00026-011-0118-1