In this note, we show that, for all $n\ge 2$, the number of distinct rooted binary phylogenetic $X$-trees displayed by a binary tree-child network $\mathcal{N}$ on $X$ with $n$ leaves is at most $2^{n-1}-1$ and that this upper bound is sharp. Furthermore, if $\mathcal{N}$ displays exactly $2^{n-1}-1$ such trees, then exactly one rooted binary phylogenetic $X$-tree is displayed twice, and this tree can be canonically found by iteratively replacing a reticulated cherry with a cherry.

翻译：在本文中，我们证明：对于所有$n\ge 2$，具有$n$片叶子的二叉树-孩子网络$\mathcal{N}$在$X$上显示的不同有根二元系统发育$X$-树的数量最多为$2^{n-1}-1$，且该上界是紧的。此外，若$\mathcal{N}$恰好显示$2^{n-1}-1$棵这样的树，则正好有一棵有根二元系统发育$X$-树被显示了两次，而这棵树可以通过迭代地将网状樱桃替换为樱桃来规范地找到。