A universal tree-based network with the minimum number of reticulations

Abstract

A tree-based network $\mathcal{N}$ on $X$ is universal if every rooted binary phylogenetic $X$-tree is a base tree for $\mathcal{N}$. Hayamizu and, independently, Zhang constructively showed that, for all positive integers $n$, there exists an universal tree-based network on $n$ leaves. For all $n$, Hayamizu’s construction contains $\Theta (n!)$ reticulations, while Zhang’s construction contains $\Theta \left(n^2\right)$ reticulations. A simple counting argument shows that a universal tree-based network has $\Omega (nlogn)$ reticulations. With this in mind, Hayamizu as well as Steel posed the problem of determining whether or not such networks exist with $O(nlogn)$ reticulations. In this paper, we show that, for all $n$, there exists a universal tree-based network on $n$ leaves with $O(nlogn)$ reticulations.