On the asymptotic joint distribution of height and width in random trees

It has been known for a long time that the height and width of a random labelled rooted tree, or of any other conditioned Galton-Watson tree, after suitable normalizations converge to the same limit distribution. Moreover, Chassaing, Marckert and Yor [7] have proved joint convergence of height and width. The resulting two-dimensional limit distribution has been studied by Donati-Martin [10]. We extend her results and give new formulas for joint moments. As an example, we calculate the covariance. We also show that the two-dimensional distribution is not symmetric, although the marginals are the same.