Open Access
2015 Height and diameter of Brownian tree
Minmin Wang
Author Affiliations +
Electron. Commun. Probab. 20: 1-15 (2015). DOI: 10.1214/ECP.v20-4193


By computations on generating functions, Szekeres proved in 1983 that the law of the diameter of a uniformly distributed rooted labelled tree with $n$ vertices, rescaled by a factor $n^{-1/2}$ , converges to a distribution whose density is explicit. Aldous observed in 1991 that this limiting distribution is the law of the diameter of the Brownian tree. In our article, we provide a computation of this law which is directly based on the normalized Brownian excursion. Moreover, we provide an explicit formula for the joint law of the height and diameter of the Brownian tree, which is a new result.


Download Citation

Minmin Wang. "Height and diameter of Brownian tree." Electron. Commun. Probab. 20 1 - 15, 2015.


Accepted: 21 November 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1335.60162
MathSciNet: MR3434205
Digital Object Identifier: 10.1214/ECP.v20-4193

Primary: 60J80

Keywords: Brownian excursion , Brownian tree , Continuum random tree , Jacobi theta function , Williams’ decomposition

Back to Top