2017/03/31 by Krzysztof Bartoszek
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Applied mathematics #Combinatorics #Computer science #Discrete mathematics #Graph theory and applications #Index (typography) #Limit (mathematics) #Mathematical analysis #Mathematics #Physics #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Tree (set theory) #math.PR #msc:05C80 #msc:60F15 #msc:60J85 #msc:62M02 #msc:62P10 #msc:92-08 #msc:92B10 #msc:92D15 #q-bio.PE #stat.AP
paper · pdf · doi:10.1016/j.mbs.2018.05.005
published as Mathematical Biosciences 303:26-45, 2018
arxiv created 2018/04/30 · openalex publication_date 2018/05/07 · arxiv updated 2020/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this work we study the limit distribution of an appropriately normalized cophenetic index of the pure-birth tree conditioned on n contemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and in L2. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction-type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure-birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time-consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.