A note on the distance in random 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 random 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 with the same probability 1/(n-1). For arbitrary positive integer i=in⩽n-1, a function of n, we demonstrate Din,n, the distance between nodes in and n in random recursive trees, is asymptotically normal as n→∞ by using the classical limit theory method.