A central limit theorem for additive functionals of increasing trees

Abstract

A tree functional is called additive if it satisfies a recursion of the form F (T ) = k j=1 F (Bj) + f (T ), where B1, . . . , B k are the branches of the tree T and f (T ) is a toll function. We prove a general central limit theorem for additive functionals of d-ary increasing trees under suitable assumptions on the toll function. The same method also applies to generalised plane-oriented increasing trees (GPORTs). One of our main applications is a log-normal law that we prove for the size of the automorphism group of d-ary increasing trees, but many other examples (old and new) are covered as well.

Reducing Simply Generated Trees by Iterative Leaf Cutting

“…There are several recent articles on properties of additive tree parameters, see for example [16], [13], and [14].…”

A Central Limit Theorem for Almost Local Additive Tree Functionals

“…It satisfies (1) with toll function f (T ) = |T | − 1, and, when suitably normalised, its limiting distribution for simply generated trees is the Airy distribution (see [15]). Previous results [5,8,14,18], while giving rather general conditions on the toll function that imply normality, are unfortunately still insufficient to cover all possible examples one might be interested in. This paper is essentially an extension of Janson's work [8] on local functionals.…”

“…Functionals of the form F S are known to be asymptotically normally distributed in different classes of trees, notably simply generated trees/Galton-Watson trees [8,18], which will also be the topic of this paper, and classes of increasing trees [5,14]. In view of this and several other important examples of additive functionals that satisfy a central limit theorem, general schemes have been devised that yield a central limit theorem under different technical assumptions.…”

“…In view of this and several other important examples of additive functionals that satisfy a central limit theorem, general schemes have been devised that yield a central limit theorem under different technical assumptions. This includes work on simply generated trees/Galton-Watson trees [8,18] (labelled trees, plane trees and d-ary trees are well-known special cases) as well as Pólya trees [18] and increasing trees [14,18] (specifically recursive trees, d-ary increasing trees and generalised plane-oriented recursive trees). It is worth mentioning, however, that there are also many instances of additive functionals that are not normally distributed in the limit, since the toll functions can be quite arbitrary.…”

“…Previous results [5,8,14,18], while giving rather general conditions on the toll function that imply normality, are unfortunately still insufficient to cover all possible examples one might be interested in. This paper is essentially an extension of Janson's work [8] on local functionals.…”