Enumerative and distributional results for d-combining tree-child networks

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 bicombining tree-child networks in which every reticulation node has exactly two parents. We extend these studies to d-combining tree-child networks where every reticulation node has now $d\ge 2$ parents, and we study one-component as well as general tree-child networks. For the number of one-component networks, we derive an exact formula from which asymptotic results follow that contain a stretched exponential for $d=2$, yet not for $d\ge 3$. For general networks, we find a novel encoding by words which leads to a recurrence for their numbers. From this recurrence, we derive asymptotic results which show the appearance of a stretched exponential for all $d\ge 2$. Moreover, we also give results on the distribution of shape parameters (e.g., number of reticulation nodes, Sackin index) of a network which is drawn uniformly at random from the set of all tree-child networks with the same number of leaves. We show phase transitions depending on d, leading to normal, Bessel, Poisson, and degenerate distributions. Some of our results are new even in the bicombining case.