Limiting Behaviour of Fréchet Means in the Space of Phylogenetic Trees

As demonstrated in Barden et al. (2013) for ${\boldsymbol T}_{\!4}$, the space of phylogenetic trees with four leaves, the global, as well as the local, topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Fr\'echet means of a probability distribution on ${\boldsymbol T}_{\!4}$. Nevertheless, the techniques used in that paper were specific to ${\boldsymbol T}_{\!4}$ and cannot be adapted to analyse Fr\'echet means in the space ${\boldsymbol T}_{\!m}$ of phylogenetic trees with $m(\geqslant5)$ leaves. To investigate the latter, this paper first studies the log map of ${\boldsymbol T}_{\!m}$, a generalisation of the inverse of the exponential map on a Riemannian manifold. Then, in terms of a modified version of the log map, we characterise Fr\'echet means in ${\boldsymbol T}_{\!m}$ and derive the limiting distributions for sample Fr\'echet means, generalising the results in Barden et al. (2013). In particular, the results show that, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Fr\'echet means lie.