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A tree-based network $\mathcal N$ on $X$ is universal if every rooted binaryphylogenetic $X$-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'sconstruction contains $\Theta(n^2)$ reticulations. A simple counting argumentshows 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 determiningwhether or not such networks exists with $O(n\log n)$ reticulations. In thispaper, we show that, for all $n$, there exists an universal tree-based networkon $n$ leaves with $O(n\log n)$ reticulations.

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