The Number of Rooted Trees of Given Depth

Abstract

In this paper it is shown that the logarithm of the number of non-isomorphic rooted trees of depth $k\geq 3$ is asymptotically $\frac{\pi^2}{6}\cdot\frac{n}{\log\log\dots\log n}$, where $\log$ is iterated $k-2$ times in the denominator.