Labelled histories with multifurcation and simultaneity

In mathematical models of phylogenetic trees evolving in time, a labelled history for a rooted labelled bifurcating tree is a temporal sequence of the branchings that give rise to the tree. That is, given a leaf-labelled tree with [Formula: see text] leaves and [Formula: see text] internal nodes, a labelled history is an identification between the internal nodes and the set [Formula: see text], such that the label assigned to a given node is strictly greater than the labels assigned to its descendants. We generalize the concept of labelled histories to [Formula: see text]-furcating trees. Consider a rooted labelled tree in which each internal node has exactly [Formula: see text] children, [Formula: see text]. We first generalize the enumeration of labelled histories for a bifurcating tree ([Formula: see text]) to enumerate labelled histories for an [Formula: see text]-furcating tree with arbitrary [Formula: see text]. We formulate a conjecture for the rooted unlabelled [Formula: see text]-furcating tree shape on [Formula: see text] internal nodes whose labelled topologies have the most labelled histories. Finally, we enumerate labelled histories for [Formula: see text]-furcating trees in a setting that allows for simultaneous branchings. These results advance mathematical phylogenetic modelling by extending computations concerning fundamental features of bifurcating phylogenetic trees to a more general class of multifurcating trees.This article is part of the theme issue '"A mathematical theory of evolution": phylogenetic models dating back 100 years'.