  The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in
Euclidean space, but the extrinsic Euclidean mean often lies outside of
treespace. Sturm showed that the intrinsic Frechet mean exists and is unique in
treespace. This Frechet mean can be approximated with an iterative algorithm,
but bounds on the convergence of the algorithm are not known, and there is no
other known polynomial algorithm for computing the Frechet mean nor even the
edges present in the mean. We give the first necessary and sufficient
conditions for an edge to be in the Frechet mean. The conditions are in the
form of inequalities on the weights of the edges. These conditions provide a
pre-processing step for finding the treespace orthant containing the Frechet
mean. This work generalizes to orthant spaces.
