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

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 $\Theta(n^2)$ reticulations. A simple counting argument shows that an universal tree-based network has $\Omega(n\log n)$ reticulations. With this in mind, Hayamizu as well as Steel posed the problem of determining whether or not such networks exists with $O(n\log n)$ reticulations. In this paper, we show that, for all $n$, there exists an universal tree-based network on $n$ leaves with $O(n\log n)$ reticulations.