Open Access
2012 Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation
Svante Janson
Probab. Surveys 9: 103-252 (2012). DOI: 10.1214/11-PS188


We give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton–Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton–Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton–Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree.

The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding theorems for this model.

This survey paper contains many known results from many different sources, together with some new results.


Download Citation

Svante Janson. "Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation." Probab. Surveys 9 103 - 252, 2012.


Published: 2012
First available in Project Euclid: 8 March 2012

zbMATH: 1244.60013
MathSciNet: MR2908619
Digital Object Identifier: 10.1214/11-PS188

Primary: 60C50
Secondary: 05C05 , 60F05 , 60J80

Keywords: balls in boxes , Galton–Watson trees , random allocations , random forests , Random trees , Simply generated trees , Size-biased Galton–Watson tree

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.9 • 2012
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