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