  Balanced minimum evolution (BME) is a statistically consistent distance-based
method to reconstruct a phylogenetic tree from an alignment of molecular data.
In 2000, Pauplin showed that the BME method is equivalent to optimizing a
linear functional over the BME polytope, the convex hull of the BME vectors
obtained from Pauplin's formula applied to all binary trees. The BME method is
related to the Neighbor Joining (NJ) algorithm, now known to be a greedy
optimization of the BME principle. Further, the NJ and BME algorithms have been
studied previously to understand when the NJ Algorithm returns a BME tree for
small numbers of taxa. In this paper we aim to elucidate the structure of the
BME polytope and strengthen knowledge of the connection between the BME method
and NJ Algorithm. We first prove that any subtree-prune-regraft move from a
binary tree to another binary tree corresponds to an edge of the BME polytope.
Moreover, we describe an entire family of faces parametrized by disjoint
clades. We show that these {\em clade-faces} are smaller dimensional BME
polytopes themselves. Finally, we show that for any order of joining nodes to
form a tree, there exists an associated distance matrix (i.e., dissimilarity
map) for which the NJ Algorithm returns the BME tree. More strongly, we show
that the BME cone and every NJ cone associated to a tree $T$ have an
intersection of positive measure.
