  As demonstrated in our previous work on ${\boldsymbol T}_{4}$, 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 ${\boldsymbol T}_{m}$ that
lie in top-dimensional or co-dimension one strata. We derive the limiting
distributions for the corresponding sample Fr\'echet means, generalising our
previous results. 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.
