  Maximum likelihood estimators are used extensively to estimate unknown
parameters of stochastic trait evolution models on phylogenetic trees. Although
the MLE has been proven to converge to the true value in the independent-sample
case, we cannot appeal to this result because trait values of different species
are correlated due to shared evolutionary history. In this paper, we consider a
$2$-state symmetric model for a single binary trait and investigate the
theoretical properties of the MLE for the transition rate in the large-tree
limit. Here, the large-tree limit is a theoretical scenario where the number of
taxa increases to infinity and we can observe the trait values for all species.
Specifically, we prove that the MLE converges to the true value under some
regularity conditions. These conditions ensure that the tree shape is not too
irregular, and holds for many practical scenarios such as trees with bounded
edges, trees generated from the Yule (pure birth) process, and trees generated
from the coalescent point process. Our result also provides an upper bound for
the distance between the MLE and the true value.
