  Galled trees are studied as a recombination model in theoretic population
genetics. This class of phylogenetic networks has been generalized to
tree-child networks, normal networks and tree-based networks by relaxing a
structural condition. Although these networks are simple, their topological
structures have yet to be fully understood. It is well-known that all
phylogenetic trees on $n$ taxa can be generated by the insertion of the $n$-th
taxa to each edge of all the phylogenetic trees on $n-1$ taxa. We prove that
all tree-child networks with $k$ reticulate nodes on $n$ taxa can be uniquely
generated via three operations from all the tree-child networks with $k-1$ or
$k$ reticulate nodes on $n-1$ taxa . An application of this result is found in
counting tree-child networks and normal networks. In particular, a simple
formula is given for the number of rooted phylogenetic networks with one
reticulate node.
