Depth of vertices with high degree in random recursive trees

Let Tn be a random recursive tree with n nodes. List vertices of Tn in decreasing order of degree as v1, . . . , vn, and write di and hi for the degree of vi and the distance of vi from the root, respectively. We prove that, as n→∞ along suitable subsequences, ( d − blog2 nc, hi − μ lnn √ σ2 lnn ) → ((Pi, i ≥ 1), (Ni, i ≥ 1)) , where μ = 1 − (log2 e)/2, σ2 = 1 − (log2 e)/4, (Pi, i ≥ 1) is a Poisson point process on Z and (Ni, i ≥ 1) is a vector of independent standard Gaussians. We additionally establish joint normality for the depths of uniformly random vertices in Tn, which extends results from [8, 14]. The joint holds even if the random vertices are conditioned to have large degree, provided the normalization is adjusted accordingly. Our results are based on a simple relationship between random recursive trees and Kingman’s n-coalescent; a utility that seems to have been largely overlooked.