Labeled histories and maximally probable labeled topologies with multifurcation

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 $n\ge 2$ 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 $n\ge 2$ 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.