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MSets of Double and Triple Weights of Trees
Elena Rubei
Dipartimento di Matematica “U. Dini”, viale Morgagni 67/A, 50134 Firenze, Italia
rubei@math.unifi.it
Annals of Combinatorics 15 (4) pp.723-734 December, 2011
AMS Subject Classification: 05C05, 05C12, 92B05
Abstract:
Let T be a weighted tree with n leaves numbered by the set {1, . . . , n}. Let Di, j(T) be the distance between the leaves i and j. Let Di, j, k(T) = 1/2( Di, j(T) +Dj, k(T) +Di, 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 Di, j.
Keywords: trees, weights of trees, neighbour-joining algorithm

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