A Sharp Lower Bound for the Number of Phylogenetic Trees Displayed by a Tree-Child Network

A normal (phylogenetic) network with k reticulations displays 2k phylogenetic trees. In this paper, we establish an analogous result for tree-child (phylogenetic) networks with no underlying 3-cycles. In particular, we show that a tree-child network with k≥2 reticulations and no underlying 3-cycles displays at least 2k/2 phylogenetic trees if k is even and at least 3222k/2 if k is odd. Moreover, we show that these bounds are sharp and characterise the tree-child networks that attain these bounds.