Labeled histories and maximally probable labeled topologies with multifurcation
Emily H. Dickey, Noah A. Rosenberg
TL;DR
This work extends labeled-history enumeration from strictly bifurcating trees to at-most-$r$-furcating trees, first without and then with simultaneous branching. It derives exact counts for total labeled histories $A_r(n)$ and for histories on a fixed topology $N(T)$, showing that the most-probable unlabeled at-most-$r$-furcating topology is strictly bifurcating, via pendant-pruning and bifurcatization transformations that preserve or increase the history count. When simultaneity is allowed, analogous totals $S_r(n)$ and topology-specific counts $E(T,z)$ are derived, and the maximization again reduces to the bifurcating case; the authors conjecture that the unique maximizing bifurcating unlabeled topology (as given by the Hammersley construction) governs the at-most-$r$-furcating case across all $n$, with some non-uniqueness possibilities for specific $(n,z)$. The results connect phylogenetic multifurcation to scheduling and precedence-constrained problems, and they solidify links to well-known combinatorial sequences, while highlighting open conjectures on uniqueness of the maximally probable topology. These contributions advance the mathematical understanding of multifurcating and temporally constrained evolutionary models and provide exact enumerative tools for related applications.
Abstract
In mathematical phylogenetics, labeled histories describe the sequences by which sets of labeled lineages coalesce to a shared ancestral lineage. We study labeled histories for at-most-$r$-furcating trees. Consider a rooted leaf-labeled tree in which internal nodes each have $i$ offspring, and $i$ is permitted to range from 2 to $r$ across internal nodes, for a specified value of $r$. For labeled topologies with $n$ leaves, we enumerate the total number of labeled histories with at-most-$r$-furcation. We enumerate the labeled histories possessed by a specific at-most-$r$-furcating labeled topology. We then demonstrate that the maximally probable at-most-$r$-furcating unlabeled topology on $n \geq 2$ leaves -- the unlabeled topology whose labelings have the largest number of labeled histories -- is the maximally probable strictly bifurcating unlabeled topology on $n$ leaves. Finally, we enumerate labeled histories for at-most-$r$-furcating labeled topologies in a setting that permits simultaneous branchings. We similarly reduce the problem of identifying the maximally probable at-most-$r$-furcating unlabeled topology on $n \geq 2$ leaves, allowing simultaneity, to that of identifying the maximally probable strictly bifurcating unlabeled topology on $n$ leaves, with simultaneity; we conjecture the shape of this bifurcating unlabeled topology. The computations contribute to the study of multifurcation, which arises in various biological processes, and they connect to analogous mathematical settings involving precedence-constrained scheduling.