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Let $T_n$ be a random recursive tree with $n$ nodes. List vertices of $T_n$in decreasing order of degree as $v^1,\ldots,v^n$, and write $d^i$ and $h^i$for the degree of $v^i$ and the distance of $v^i$ from the root, respectively.We prove that, as $n \to \infty$ along suitable subsequences, \[ \bigg(d^i -\lfloor \log_2 n \rfloor, \frac{h^i - \mu\ln n}{\sqrt{\sigma^2\ln n}}\bigg) \to((P_i,i \ge 1),(N_i,i \ge 1))\, , \] where $\mu=1-(\log_2 e)/2$,$\sigma^2=1-(\log_2 e)/4$, $(P_i,i \ge 1)$ is a Poisson point process on$\mathbb{Z}$ and $(N_i,i \ge 1)$ is a vector of independent standard Gaussians.We additionally establish joint normality for the depths of uniformly randomvertices in $T_n$, which extends results for the case of a single randomvertex. The joint limit holds even if the random vertices are conditioned tohave large degree, provided the normalizing constants are adjusted accordingly;however, both the mean and variance of the conditinal depths remain of orden$\ln n$.  Our results are based on a simple relationship between random recursive treesand Kingman's $n$-coalescent; a utility that seems to have been largelyoverlooked.

Depth of vertices with high degree in random recursive trees