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X-Trees and Weighted Quartet Systems
Andreas W.M. Dress1 and Péter L. Erdős2
1Forschungsschwerpunkt Mathematisierung-Struktubildugprozesse, University of Bielefeld P.O. Box 100131, 33501 Bielefeld, Germany
dress@mathematik.uni-bielefeld.de
2A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O. Box 127 1364 Hungary
elp@renyi.hu
Annals of Combinatorics 7 (2) p.155-169 June, 2003
AMS Subject Classification: 05C05, 92D15, 92B05
Abstract:
In this note, we consider a finite set X and maps W from the set of all 2, 2-splits of X into . We show that such a map W is induced, in a canonical way, by a binary X-tree for which a positive length is associated to every inner edge e if and only if (i) exactly two of the three numbers W(ab|cd), W(ac|bd), and W(ad|cb) vanish, for any four distinct elements a, b, c, d in X, (ii) a d and W(ab|xc) + W(ax|cd) = W(ab|cd) holds for all a, b, c, d, x in X with #{a, b, c, x} = #{b, c, d, x}=4 and W(ab|cx), W(ax|cd) > 0 and (iii) W(ab|uv) min(W(ab|uw), W(ab|vw)) holds for any five distinct elements a, b, u, v, w in X. Possible generalizations regarding arbitrary R-trees and applications regarding tree-reconstruction algorithms are indicated.
Keywords: biological systematics, phylogeny, phylogenetic combinatorics, evolutionary trees, tree reconstruction, X-trees, quartet methods, quartet systems, weighted quartet systems.

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