Fringe subtrees of split trees and fractional split trees

TL;DR

This paper studies additive functionals on split and fractional split trees, proving convergence of moments, distribution limits, and laws of large numbers, with explicit formulas and approximations for various models.

Contribution

It introduces fractional split trees, extends analysis of additive functionals, and derives explicit limit formulas and distribution results for these generalized tree models.

Findings

First moment converges to an explicit limit in certain models.

Standard deviation is smaller than the mean in some cases, leading to a weak law of large numbers.

Distribution limits are established using the contraction method in specific fractional split trees.

Abstract

We consider additive functionals $X_n(\phi)$ with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if $s_0=s_1=0$ and for some models with Beta-distributed splitter. For $s_0+s_1>0$, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method.