Probability Surveys

Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation

Svante Janson

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Abstract

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.

Article information

Source
Probab. Surveys, Volume 9 (2012), 103-252.

Dates
First available in Project Euclid: 8 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.ps/1331216239

Digital Object Identifier
doi:10.1214/11-PS188

Mathematical Reviews number (MathSciNet)
MR2908619

Zentralblatt MATH identifier
1244.60013

Subjects
Primary: 60C50
Secondary: 05C05: Trees 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Random trees simply generated trees Galton–Watson trees random allocations balls in boxes size-biased Galton–Watson tree random forests

Citation

Janson, Svante. Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surveys 9 (2012), 103--252. doi:10.1214/11-PS188. https://projecteuclid.org/euclid.ps/1331216239


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