  Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive
tree with $n+1$ vertices. We prove a functional limit theorem for the
vector-valued process $(X_{[n^t]}(1),\ldots, X_{[n^t]}(k))_{t\geq 0}$, for each
$k\in\mathbb N$. We show that after proper centering and normalization, this
process converges weakly to a vector-valued Gaussian process whose components
are integrated Brownian motions. This result is deduced from a functional limit
theorem for Crump-Mode-Jagers branching processes generated by increasing
random walks with increments that have finite second moment.
