On the distribution of distances in recursive trees

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree Tn with n labeled nodes is a recursive tree if n = 1, or n > 1 and Tn can be constructed by joining node n to a node of some recursive tree Tn−1. For arbitrary nodes i < n in a random recursive tree we give the exact distribution of Xi,n, the distance between nodes i and n. We characterize this distribution as the convolution of the law of Xi,i+1 and n−i−1 Bernoulli distributions. We further characterize the law of Xi,i+1 as a mixture of sums of Bernoullis. For i = in growing as a function of n, we show that Xin,n is asymptotically normal in several settings. AMS 1991 subject classifications. Primary 05C05; secondary 60C05