- Probab. Surveys
- Volume 9 (2012), 103-252.
Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation
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.
Probab. Surveys, Volume 9 (2012), 103-252.
First available in Project Euclid: 8 March 2012
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Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Secondary: 05C05: Trees 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
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