  The popular neighbor-joining (NJ) algorithm used in phylogenetics is a greedy
algorithm for finding the balanced minimum evolution (BME) tree associated to a
dissimilarity map. From this point of view, NJ is ``optimal'' when the
algorithm outputs the tree which minimizes the balanced minimum evolution
criterion. We use the fact that the NJ tree topology and the BME tree topology
are determined by polyhedral subdivisions of the spaces of dissimilarity maps
${\R}_{+}^{n \choose 2}$ to study the optimality of the neighbor-joining
algorithm. In particular, we investigate and compare the polyhedral
subdivisions for $n \leq 8$. A key requirement is the measurement of volumes of
spherical polytopes in high dimension, which we obtain using a combination of
Monte Carlo methods and polyhedral algorithms. We show that highly unrelated
trees can be co-optimal in BME reconstruction, and that NJ regions are not
convex. We obtain the $l_2$ radius for neighbor-joining for $n=5$ and we
conjecture that the ability of the neighbor-joining algorithm to recover the
BME tree depends on the diameter of the BME tree.
