| Affiliation: | (1) Forschungsschwerpunkt Mathematisierungs-Strukturbildungsprozesse, University of Bielefeld, 100131, 33501 Bielefeld, Germany;(2) A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, 127, 1364 Budapest, Hungary |
| Abstract: | In this note, we consider a finite set X and maps W from the set $ mathcal{S}_{2|2} (X) $ of all 2, 2-splits of X into $ mathbb{R}_{geq 0} $. We show that such a map W is induced, in a canonical way, by a binaryX-tree for which a positive length $ mathcal{l} (e) $ is associated to every inner edge e if and only if (i) exactlytwo of the three numbers W(ab|cd),W(ac|bd), and W(ad|cb) vanish, for any four distinct elementsa, b, c, d in X, (ii) $ a neq d quadmathrm{and}quad W (ab|xc) + W(ax|cd) = W(ab|cd) $ holds for all a, b, c, d, xin X with #{a, b, c, x} = #{b, c, d, x} = 4 and $ W(ab|cx),W(ax|cd) $ > 0, and (iii) $ W (ab|uv) geq quad mathrm{min} (W(ab|uw), W(ab|vw)) $holds for any five distinct elements a, b, u, v, w in X. Possible generalizationsregarding arbitrary $ mathbb{R} $-trees and applications regarding tree-reconstruction algorithmsare indicated.AMS Subject Classification: 05C05, 92D15, 92B05. |
