We address an open question of Francis and Steel about phylogenetic networks and trees. They give a polynomial time algorithm to decide if a phylogenetic network, N , is tree-based and pose the problem: given a fixed tree T and network N , is N based on T ? We show that it is N P -hard to decide, by reduction from 3-Dimensional Matching (3DM), and further, that the problem is fixed parameter tractable.

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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 Fréchet mean exists and is unique in treespace. This Fréchet 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 Fréchet mean nor even the edges present in the mean. We give the first necessary and sufficient conditions for an edge to be in the Fréchet 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 Fréchet mean. This work generalizes to orthant spaces.