Probability Surveys

Random trees and applications

Jean-François Le Gall

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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.

Article information

Probab. Surveys, Volume 2 (2005), 245-311.

First available in Project Euclid: 21 November 2005

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 05C05: Trees 35J65: Nonlinear boundary value problems for linear elliptic equations 60C05: Combinatorial probability 60J65: Brownian motion [See also 58J65]

random tree contour process conditioned tree Brownian motion Brownian excursion real tree coding of trees CRT Brownian snake exit measure partial differential equation


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

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