On the distribution of the number of vertices in strata of a random tree

The set of trees with $m+1$ distinguishable vertices one of which is taken as a root is considered. In every tree all the vertices are distributed in strata with respect to the root according to the lengths of paths which connect them to the root.
Let $\zeta_{m,j}$ be the number of vertices in the $j$-th stratum of a tree chosen at random.
We prove that if $m$ and $j\to\infty$ so that $j/\sqrt m\to\alpha$, $0<\alpha_1\le\alpha<\alpha_2<\infty$, then the distributions of random variables $\zeta_{m,j}/\sqrt m$ converge to a limit distribution. Explicit expressions for moments and the density of the limit distribution are found.