  Orchard and tree-child networks share an important property with phylogenetic
trees: they can be completely reduced to a single node by iteratively deleting
cherries and reticulated cherries. As it is the case with phylogenetic trees,
the number of ways in which this can be done gives information about the
topology of the network. Here, we show that the problem of computing this
number in tree-child networks is akin to that of finding the number of linear
extensions of the poset induced by each network, and give an algorithm based on
this reduction whose complexity is bounded in terms of the level of the
network.
