Abstract
We give a unified treatment of the limit, as the size tends to infinity, of random simply generated 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 random infinite limit tree can in all cases be constructed by a modified Galton-Watson process. In the standard case of a critical Galton-Watson tree, the limit tree has an infinite "spine", where the offspring distribution is size-biased. In the other cases, the spine has finite length and ends with a vertex with infinite degree. A node of infinite degree in the limit corresponds to the existence of one node with very high degree in the finite random trees; in physics terminology, this is a type of condensation. In simple cases, there is one node with a degree that is roughly a constant times the number of nodes, while all other degrees are much smaller; however, more complicated behaviour is also possible. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding results for this model.
Highlights
We give a unified treatment of the limit, as the size tends to infinity, of random generated 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
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 random infinite limit tree can in all cases be constructed by a modified Galton-Watson process
Summary
(We say that (wk) is a probability weight sequence.) In this case we let ξ be a random variable with the corresponding distribution: P(ξ = k) = wk. In many cases it is possible to change the weight (wk) to an equivalent probability weight sequence; in this case Tn can be seen as a conditioned Galton–Watson tree. In many cases this can be done such that the resulting probability distribution has mean 1 In such cases it suffices to consider the case of a probability weight sequence with mean E ξ = 1; Tn is a conditional critical Galton–Watson tree. We extend here some of these results to the general case, including cases where no equivalent probability weight sequence exists
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