Convergence Rate for The Number of Crossing in a Random Labelled Tree

We consider the number of crossings in a random labelled tree with vertices in convex position. We give a new proof of the fact that this quantity is asymptotically Gaussian with mean $n^2\mathrm{/}6$ and variance $n^3\mathrm{/}45$. Furthermore, we give an estimate for the Kolmogorov distance to a Gaussian distribution which implies a convergence rate of order $n^{-1\mathrm{/}2}$.