  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 random
vertices in $T_n$, which extends results for the case of a single random
vertex. The joint limit holds even if the random vertices are conditioned to
have 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 trees
and Kingman's $n$-coalescent; a utility that seems to have been largely
overlooked.
