Limiting Behaviour of Fréchet Means in the Space of Phylogenetic Trees

This paper continues the investigation initiated in Barden et al. (2013), studying the characterisation, and limiting distributions, of Fr\'echet means in the space ${\boldsymbol T}_{\!m}$ of phylogenetic trees with $m(\geqslant3)$ leaves. We generalise the results that we obtained in Barden et al. (2013) for ${\boldsymbol T}_{\!4}$ and demonstrate the non-classical behaviour of Fr\'echet means arising from the global topological structure of the space. In particular, as in Barden et al. (2013), we 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.