Insertion depth in power-weight trees

Abstract

We study the insertion depth in a class of nonuniform random recursive trees grown with an attachment preference for a power of the node index. The strength of index preference is controlled by a real-valued power parameter α; the model accommodates both young-age and old-age preference as specific cases. We find the exact probability law in terms of the Poisson-Binomial distribution, and consequently, the exact and asymptotic mean and variance. Under appropriate normalization, we derive concentration laws and limiting distributions. For α>−1, with logarithmic normalization of the depth, we have a normal limit. The case α=−1 is a critical point at which we retain a normal limit but under an iterated logarithm normalization. For these normal cases, Chen-Stein approximation gives a slow rate of convergence in the Wasserstein distance. The case α<−1 is a phase that has a different behavior.