A sharp lower bound for the number of phylogenetic trees displayed by a tree-child network

A normal (phylogenetic) network with $k$ reticulations displays $2^k$ 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 $k\ge 2$ reticulations and no underlying $3$-cycles displays at least $2^{k\mathrm{/}2}$ phylogenetic trees if $k$ is even and at least $\frac{3}{2\sqrt{2}}{2}^{k\mathrm{/}2}$ if $k$ is odd. Moreover, we show that these bounds are sharp and characterise the tree-child networks that attain these bounds.