New from @alexei_drummond and his postdoc:
The space of ultrametric phylogenetic trees by Alex Gavruskin, Alexei J.
We introduce two metric spaces on ultrametric phylogenetic trees and
compare them with existing models of tree space. We formulate several
formal requirements that a metric space on phylogenetic trees must
possess in order to be a satisfactory space for statistical analysis,
and justify them. We show that only a few known constructions of the
space of phylogenetic trees satisfy these requirements. However, our
results suggest that these basic requirements are not enough to
distinguish between the two metric spaces we introduce and that the
choice between metric spaces requires additional properties to be
I haven't read it in detail, but it seems that the most version of the space that is most natural for time-trees (their t-space) has properties that make it mathematically difficult to analyze. The combinatorial machinery that helped out with the BHV space doesn't help here.
Theorem 8. The problem of computing geodesics in t-space is NP-hard.
We will reduce the problem of computing NNI-distance to the problem of
computing geodesics in t-space, but before going on to the proof of
this result, we would like to develop some intuition of why t-space is
so different from both BHV and τ -space. The key property for this
difference is that the cone-path is rarely a geodesic in t-space.
Indeed, in both BHV and τ - space the position of two cubes can result
in a cone-path being the geodesic between any pair of trees from these
cubes. Particularly, the measure of the set of pairs of trees between
which the cone-path is a geodesic is positive. For example, if two
trees T and R have topologies with no compatible splits, then the
geodesic between T and R is a cone-path . A property such as this
does not present in t-space. It will follow from the observations
below that the measure of the set of pairs of trees between which the
geodesic is a cone-path in t-space has measure 0.
I know @cwhidden has been reading it so perhaps he'll post some observations.