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

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