A Combinatorial Approach for Analyzing the Number of Descendants in Increasing Trees and Related Parameters

This work is devoted to a study of the number of descendants of node j in random increasing trees, previously treated in [4, 6, 8, 13], and also to a study of a more general quantity called generalized descendants, which arises in a certain growth process. Our study is based on a combinatorial approach by establishing a bijection with certain lattice paths. For the parameters considered we derive closed formulæ for the probability distributions, the expectation and the variance, and obtain limiting distribution results also, extending known results in the literature. Furthermore, the bijective approach enables us to study a weighted version of the number of descendants of node j in random increasing trees. Moreover, we also discuss the multidimensional case, i.e., the joint distribution of the number of descendants of the nodes j1 and j2, and applications.