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 Probability Surveys > Vol. 9 (2012) open journal systems 

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

Svante Janson, Uppsala University

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.

AMS 2000 subject classifications: Primary 60C50; secondary 05C05, 60F05, 60J80.

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

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


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