Galled Tree-Child Networks

TL;DR

This paper introduces galled tree-child networks, a new class combining features of galled and tree-child networks, and analyzes their counting and statistical properties, revealing asymptotic normality in reticulation nodes.

Contribution

It defines the novel class of galled tree-child networks and extends counting and stochastic analysis methods to this class, showing asymptotic normality of reticulation nodes.

Findings

Counting results for galled tree-child networks follow similar methods as galled networks.

Number of reticulation nodes is asymptotically normally distributed.

Contrasts with the discrete limit laws of galled and tree-child networks.

Abstract

We propose the class of galled tree-child networks which is obtained as intersection of the classes of galled networks and tree-child networks. For the latter two classes, (asymptotic) counting results and stochastic results have been proved with very different methods. We show that a counting result for the class of galled tree-child networks follows with similar tools as used for galled networks, however, the result has a similar pattern as the one for tree-child networks. In addition, we also consider the (suitably scaled) numbers of reticulation nodes of random galled tree-child networks and show that they are asymptotically normal distributed. This is in contrast to the limit laws of the corresponding quantities for galled networks and tree-child networks which have been both shown to be discrete.