On the distribution of distances between specified nodes in increasing trees

We study the quantity distance between node  jjand node  nnin a random tree of size  nn 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