Analysis of label-based parameters in increasing trees

Grown simple families of increasing trees are a subclass of increasing trees, which can be constructed by an insertion process. Three such tree families contained in the grown simple families of increasing trees are of particular interest: recursive trees, plane-oriented recursive trees and binary increasing trees. Here we present a general approach for the analysis of a number of label-based parameters in a random grown simple increasing tree of size n as, e.g., the degree of the node labeled j, the subtree-size of the node labeled j, etc. Further we apply the approach to the random variable Xn;j;a, which counts the number of size-a branches attached to the node labeled j ( = subtrees of size a rooted at the children of the node labeled j) in a random grown simple increasing tree of size n. We can give closed formulæ for the probability distribution and the factorial moments. Furthermore limiting distribution results for Xn;j;a are given dependent on the growth behavior of j = j(n) compared to n