### Descendants in Increasing Trees

#### Abstract

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size $n$ all its increasing labellings, i.$\,$e. labellings of the nodes by distinct integers of the set $\{1, \dots, n\}$ in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree families are of particular interest: *recursive trees*, *plane-oriented recursive trees* and *binary increasing trees*. We study the quantity *number of descendants of node $j$ in a random tree of size $n$* and give closed formulæ for the probability distribution and all factorial moments for those subclass of tree families, which can be constructed via an insertion process. Furthermore limiting distribution results of this parameter are given.