A note on the distance in random recursive trees

Abstract

Recursive trees have been used to model such things as the spread of epidemics, family trees of ancient manuscripts, and pyramid schemes. A tree T n with n labeled nodes is a random recursive tree if n = 1 , or n > 1 and T n can be constructed by joining node n to a node of some recursive tree T n - 1 with the same probability 1 / ( n - 1 ) . For arbitrary positive integer i = i n ⩽ n - 1 , a function of n, we demonstrate D i n , n , the distance between nodes i n and n in random recursive trees, is asymptotically normal as n → ∞ by using the classical limit theory method.