On the distribution of distances between specified nodes in increasing trees

We study the quantity distance between node j and node n in a random tree of size n chosen from a family of increasing trees. For those subclass of increasing tree families, which can be constructed via a tree evolution process, we give closed formulæ for the probability distribution, the expectation and the variance. Furthermore we derive a distributional decomposition of the random variable considered and we show a central limit theorem of this quantity, for arbitrary labels 1 ≤ j < n and n→∞. Such tree models are of particular interest in applications, e.g., the widely used models of recursive trees, plane-oriented recursive trees and binary increasing trees are special instances and are thus covered by our results.