When two trees go to war

Rooted phylogenetic networks are often constructed by combining trees, clusters, triplets or characters into a single network that in some well-defined sense simultaneously represents them all. We review these four models and investigate how they are related. In general, the model chosen influences the minimum number of reticulation events required. However, when one obtains the input data from two binary trees, we show that the minimum number of reticulations is independent of the model. The number of reticulations necessary to represent the trees, triplets, clusters (in the softwired sense) and characters (with unrestricted multiple crossover recombination) are all equal. Furthermore, we show that these results also hold when not the number of reticulations but the level of the constructed network is minimised. We use these unification results to settle several complexity questions that have been open in the field for some time. We also give explicit examples to show that already for data obtained from three binary trees the models begin to diverge

Kernelizations for the hybridization number problem on multiple nonbinary trees

Given a finite set $X$, a collection $\mathcal{T}$ of rooted phylogenetic trees on $X$ and an integer $k$, the Hybridization Number problem asks if there exists a phylogenetic network on $X$ that displays all trees from $\mathcal{T}$ and has reticulation number at most $k$. We show two kernelization algorithms for Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(\Delta^+-1)$ respectively, with $t$ the number of input trees and $\Delta^+$ their maximum outdegree. Experiments on simulated data demonstrate the practical relevance of these kernelization algorithms. In addition, we present an $n^{f(k)}t$-time algorithm, with $n=|X|$ and $f$ some computable function of $k$

On the enumeration of leaf-labelled increasing trees with arbitrary node-degree

We consider the counting problem of the number of \textit{leaf-labeled increasing trees}, where internal nodes may have an arbitrary number of descendants. The set of all such trees is a discrete representation of the genealogies obtained under certain population-genetical models such as multiple-merger coalescents. While the combinatorics of the binary trees among those are well understood, for the number of all trees only an approximate asymptotic formula is known. In this work, we validate this formula up to constant terms and compare the asymptotic behavior of the number of all leaf-labelled increasing trees to that of binary, ternary and quaternary trees