  Diffusion processes on trees are commonly used in evolutionary biology to
model the joint distribution of continuous traits, such as body mass, across
species. Estimating the parameters of such processes from tip values presents
challenges because of the intrinsic correlation between the observations
produced by the shared evolutionary history, thus violating the standard
independence assumption of large-sample theory. For instance Ho and An\'e
\cite{HoAne13} recently proved that the mean (also known in this context as
selection optimum) of an Ornstein-Uhlenbeck process on a tree cannot be
estimated consistently from an increasing number of tip observations if the
tree height is bounded. Here, using a fruitful connection to the so-called
reconstruction problem in probability theory, we study the convergence rate of
parameter estimation in the unbounded height case. For the mean of the process,
we provide a necessary and sufficient condition for the consistency of the
maximum likelihood estimator (MLE) and establish a phase transition on its
convergence rate in terms of the growth of the tree. In particular we show that
a loss of $\sqrt{n}$-consistency (i.e., the variance of the MLE becomes
$\Omega(n^{-1})$, where $n$ is the number of tips) occurs when the tree growth
is larger than a threshold related to the phase transition of the
reconstruction problem. For the covariance parameters, we give a novel,
efficient estimation method which achieves $\sqrt{n}$-consistency under natural
assumptions on the tree.
