When is a set of phylogenetic trees displayed by a normal network?

TL;DR

This paper develops a polynomial-time algorithm to reconstruct unique binary normal networks from a set of phylogenetic trees, characterizes when collections can be displayed by such networks, and compares their complexity to tree-child networks.

Contribution

It introduces a novel polynomial-time reconstruction algorithm for normal networks and characterizes display conditions using cherry-picking sequences.

Findings

Normal networks are uniquely determined by displayed trees.

Any two rooted phylogenetic trees can be displayed by a normal network.

Normal networks require more reticulations than tree-child networks for certain tree collections.

Abstract

A normal network is uniquely determined by the set of phylogenetic trees that it displays. Given a set $\mathcal{P}$ of rooted binary phylogenetic trees, this paper presents a polynomial-time algorithm that reconstructs the unique binary normal network whose set of displayed binary trees is $\mathcal{P}$, if such a network exists. Additionally, we show that any two rooted phylogenetic trees can be displayed by a normal network and show that this result does not extend to more than two trees. This is in contrast to tree-child networks where it has been previously shown that any collection of rooted phylogenetic trees can be displayed by a tree-child network. Lastly, we introduce a type of cherry-picking sequence that characterises when a collection $\mathcal{P}$ of rooted phylogenetic trees can be displayed by a normal network and, further, characterise the minimum number of reticulations needed over all normal networks that display $\mathcal{P}$. We then exploit these sequences to show that, for all $n\ge 3$, there exist two rooted binary phylogenetic trees on $n$ leaves that can be displayed by a tree-child network with a single reticulation, but cannot be displayed by a normal network with less than $n-2$ reticulations.