# On the number of crossings in a random labelled tree with vertices in convex position

## TL;DR

This paper proves that the number of crossings in a random labelled tree with vertices in convex position follows an asymptotically Gaussian distribution, with specific mean and variance, and extends the result to points in general position.

## Contribution

It establishes the asymptotic Gaussian distribution of crossings in random labelled trees with vertices in convex and general positions, providing new probabilistic insights.

## Key findings

- Crossings are asymptotically Gaussian distributed.
- Mean number of crossings is approximately n^2/6.
- Variance of crossings is approximately n^3/45.

## Abstract

We prove that the number of crossings in a random labelled tree with vertices in convex position is asymptotically Gaussian with mean $ n^2/6$ and variance $ n^3/45$. A similar result is proved for points in general position under mild constraints.