  In previous work, we gave asymptotic counting results for the number of
tree-child and normal networks with $k$ reticulation vertices and explicit
exponential generating functions of the counting sequences for $k=1,2,3$. The
purpose of this note is two-fold. First, we make some corrections to our
previous approach which overcounted the above numbers and thus gives erroneous
exponential generating functions (however, the overcounting does not effect our
asymptotic counting results). Secondly, we use our (corrected) exponential
generating functions to derive explicit formulas for the number of tree-child
and normal networks with $k=1,2,3$ reticulation vertices. This re-derives
recent results of Carona and Zhang, answers their question for normal networks
with $k=2$, and adds new formulas in the case $k=3$.
