  Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive
tree with $n+1$ vertices. We are interested in the asymptotic behavior of
$X_n(k)$ for intermediate levels $k=k_n$ satisfying $k_n\to\infty$ and
$k_n=o(\log n)$ as $n\to\infty$. In particular, we prove weak convergence of
finite-dimensional distributions for the process $(X_n ([k_n u]))_{u>0}$,
properly normalized and centered, as $n\to\infty$. The limit is a centered
Gaussian process with covariance $(u,v)\mapsto (u+v)^{-1}$. One-dimensional
distributional convergence of $X_n(k_n)$, properly normalized and centered, was
obtained with the help of analytic tools by Fuchs, Hwang and Neininger in 2006.
In contrast, our proofs which are probabilistic in nature exploit a connection
of our model with certain Crump-Mode-Jagers branching processes.
