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

TL;DR

This paper unifies the understanding of the asymptotic behavior of simply generated random trees, including critical and non-critical cases, revealing a common limit structure involving infinite trees and a connection to balls-in-boxes models.

Contribution

It provides a unified framework for the limit behavior of simply generated trees across different regimes, including new results and a connection to random allocation models.

Findings

Limit trees are well-defined in all cases, with critical cases yielding locally finite trees.

Non-critical cases produce infinite trees with a single node of infinite degree.

The connection to balls-in-boxes models facilitates the analysis of these limits.

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.