  Tree-child networks, one of the prominent network classes in phylogenetics,
have been introduced for the purpose of modeling reticulate evolution.
Recently, the first author together with Gittenberger and Mansouri (2019)
showed that the number ${\rm TC}_{\ell,k}$ of tree-child networks with $\ell$
leaves and $k$ reticulation vertices has the first-order asymptotics \[ {\rm
TC}_{\ell,k}\sim c_k\left(\frac{2}{e}\right)^{\ell}\ell^{\ell+2k-1},\qquad
(\ell\rightarrow\infty). \] Moreover, they also computed $c_k$ for $k=1,2,$ and
$3$. In this short note, we give a second approach to the above result which is
based on a recent (algorithmic) approach for the counting of tree-child
networks due to Cardona and Zhang (2020). This second approach is also capable
of giving a simple, closed-form expression for $c_k$ for all $k\geq 0$.
