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

• Published in: Journal of computational biology
• Main Authors: Fuchs, Michael, Steel, Mike
• Format: Journal Article
• Language: English
• Published: United States Mary Ann Liebert, Inc., publishers 02.07.2025
• Subjects: Poisson and Normal distributions; RF dissimilarity metric; asymptotic enumeration; Cayley trees; k-RF dissimilarity metric
• ISSN: 1557-8666; 1557-8666
• DOI: 10.1089/cmb.2025.0093

Motivated by applications in medical bioinformatics, Khayatian et al. (2024) introduced a family of metrics on Cayley trees [the k -Robinson–Foulds (RF) distance, for k   =   0 ,  . . .  , n − 2 ] 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 k -RF metric for the extreme values k   =   0 and k   =   n − 2 , as n 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 ( n − 2 )-RF metric leads to a normal distribution (with mean ∼   n e − 2 ). These results (together with the case k   =   1 which behaves quite differently and k   =   n − 3 ) shed light on the earlier simulation results and the predictions made concerning them.