  Given a gene-tree labeled topology $G$ and a species tree $S$, the "ancestral
configurations" at an internal node $k$ of $S$ represent the combinatorially
different sets of gene lineages that can be present at $k$ when all possible
realizations of $G$ in $S$ are considered. Ancestral configurations have been
introduced as a data structure for evaluating the conditional probability of a
gene-tree labeled topology given a species tree, and their enumeration assists
in describing the complexity of this computation. In the case that the
gene-tree labeled topology $G=t$ matches that of the species tree $S$, by
techniques of analytic combinatorics, we study distributional properties of the
"total" number of ancestral configurations measured across the different nodes
of a random labeled topology $t$ selected under the uniform and the Yule
probability models. Under both of these probabilistic scenarios, we show that
the total number $T_n$ of ancestral configurations of a random labeled topology
of $n$ taxa asymptotically follows a lognormal distribution. Over uniformly
distributed labeled topologies, the asymptotic growth of the mean and the
variance of $T_n$ are found to satisfy $\mathbb{E}_{\rm U}[T_n] \sim 2.449
\cdot 1.333^n$ and $\mathbb{V}_{\rm U}[T_n] \sim 5.050 \cdot 1.822^n$,
respectively. Under the Yule model, which assigns higher probabilities to more
balanced labeled topologies, we obtain the mean $\mathbb{E}_{\rm Y}[T_n] \sim
1.425^n$ and the variance $\mathbb{V}_{\rm Y}[T_n] \sim 2.045^n$.
