Profile of Random Exponential Recursive Trees

Abstract

We introduce the random exponential recursive tree in which at each point of discrete time every node recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We study the distribution of the size of these trees and the average level composition, often called the profile. We also study the size and profile of an exponential version of the plane-oriented recursive tree (PORT), wherein every insertion positions in the “gaps” between the edges recruits a child (new leaf) with probability p, or fails to do so with probability 1 − p. We use martingales in conjunction with distributional equations to establish strong laws for the size of both exponential flavors; in both cases, the limit laws are characterized by their moments. Via generating functions, we get an exact expression for the average expectation of the number of nodes at each level. Asymptotic analysis reveals that the most populous level is $\frac {p}{p+1} n$ in exponential recursive trees, and is $\frac {p}{2p+1} n$ in exponential PORTs.