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Distance-based phylogenetic algorithms attempt to solve the NP-hard leastsquares phylogeny problem by mapping an arbitrary dissimilarity maprepresenting biological data to a tree metric. The set of all dissimilaritymaps is a Euclidean space properly containing the space of all tree metrics asa polyhedral fan. Outputs of distance-based tree reconstruction algorithms suchas UPGMA and Neighbor-Joining are points in the maximal cones in the fan. Treemetrics with polytomies lie at the intersections of maximal cones.  A phylogenetic algorithm divides the space of all dissimilarity maps intoregions based upon which combinatorial tree is reconstructed by the algorithm.Comparison of phylogenetic methods can be done by comparing the geometry ofthese regions. We use polyhedral geometry to compare the local nature of thesubdivisions induced by least squares phylogeny, UPGMA, and Neighbor-Joining.Our results suggest that in some circumstances, UPGMA and Neighbor-Joiningpoorly match least squares phylogeny when the true tree has a polytomy.

Distance-based phylogenetic methods around a polytomy