Open Access
2005 Random trees and applications
Jean-François Le Gall
Probab. Surveys 2: 245-311 (2005). DOI: 10.1214/154957805100000140


We discuss several connections between discrete and continuous random trees. In the discrete setting, we focus on Galton-Watson trees under various conditionings. In particular, we present a simple approach to Aldous’ theorem giving the convergence in distribution of the contour process of conditioned Galton-Watson trees towards the normalized Brownian excursion. We also briefly discuss applications to combinatorial trees. In the continuous setting, we use the formalism of real trees, which yields an elegant formulation of the convergence of rescaled discrete trees towards continuous objects. We explain the coding of real trees by functions, which is a continuous version of the well-known coding of discrete trees by Dyck paths. We pay special attention to random real trees coded by Brownian excursions, and in a particular we provide a simple derivation of the marginal distributions of the CRT. The last section is an introduction to the theory of the Brownian snake, which combines the genealogical structure of random real trees with independent spatial motions. We introduce exit measures for the Brownian snake and we present some applications to a class of semilinear partial differential equations.


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Jean-François Le Gall. "Random trees and applications." Probab. Surveys 2 245 - 311, 2005.


Published: 2005
First available in Project Euclid: 21 November 2005

zbMATH: 1189.60161
MathSciNet: MR2203728
Digital Object Identifier: 10.1214/154957805100000140

Primary: 60J80
Secondary: 05C05 , 35J65 , 60C05 , 60J65

Keywords: Brownian excursion , Brownian motion , Brownian snake , coding of trees , conditioned tree , contour process , CRT , Exit measure , partial differential equation , Random tree , real tree

Rights: Copyright © 2005 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.2 • 2005
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