The Asymptotic Distribution of the -Robinson-Foulds Dissimilarity Measure on Labeled Trees.

Motivated by applications in medical bioinformatics, Khayatian et al. (2024) introduced a family of metrics on Cayley trees [the -Robinson-Foulds (RF) distance, for  . . . ] and explored their distribution on pairs of random Cayley trees via simulations. In this article, we investigate this distribution mathematically and derive exact asymptotic descriptions of the distribution of the -RF metric for the extreme values and , as becomes large. We show that a linear transform of the 0-RF metric converges to a Poisson distribution (with mean 2), whereas a similar transform for the ()-RF metric leads to a normal distribution (with mean ). These results (together with the case which behaves quite differently and ) shed light on the earlier simulation results and the predictions made concerning them.