  Tree-child networks are one of the most prominent network classes for
modeling evolutionary processes which contain reticulation events. Several
recent studies have addressed counting questions for {\it bicombining
tree-child networks} which are tree-child networks with every reticulation node
having exactly two parents. In this paper, we extend these studies to {\it
$d$-combining tree-child networks} where every reticulation node has now $d\geq
2$ parents. Moreover, we also give results and conjectures on the
distributional behavior of the number of reticulation nodes of a network which
is drawn uniformly at random from the set of all tree-child networks with the
same number of leaves.
