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Given a gene-tree labeled topology $G$ and a species tree $S$, the "ancestralconfigurations" at an internal node $k$ of $S$ represent the combinatoriallydifferent sets of gene lineages that can be present at $k$ when all possiblerealizations of $G$ in $S$ are considered. Ancestral configurations have beenintroduced as a data structure for evaluating the conditional probability of agene-tree labeled topology given a species tree, and their enumeration assistsin describing the complexity of this computation. In the case that thegene-tree labeled topology $G=t$ matches that of the species tree $S$, bytechniques of analytic combinatorics, we study distributional properties of the"total" number of ancestral configurations measured across the different nodesof a random labeled topology $t$ selected under the uniform and the Yuleprobability models. Under both of these probabilistic scenarios, we show thatthe total number $T_n$ of ancestral configurations of a random labeled topologyof $n$ taxa asymptotically follows a lognormal distribution. Over uniformlydistributed labeled topologies, the asymptotic growth of the mean and thevariance 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 morebalanced labeled topologies, we obtain the mean $\mathbb{E}_{\rm Y}[T_n] \sim1.425^n$ and the variance $\mathbb{V}_{\rm Y}[T_n] \sim 2.045^n$.

The distributions under two species-tree models of the total number of ancestral configurations for matching gene trees and species trees