Some remarks on biased recursive trees

TL;DR

This paper analyzes biased recursive trees, a non-uniform random tree model linked to biased permutations, providing new insights into tree statistics and their limits, including connections to uniform recursive trees.

Contribution

It introduces and analyzes biased recursive trees, revealing new statistical properties and their relation to uniform recursive trees as a special case.

Findings

Derived formulas for node descendants and tree depth.

Established the limit to uniform recursive trees.

Provided new results for uniform recursive trees.

Abstract

The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform permutations, biased riffle shuffles. The statistics that are treated include the number of nodes with a given number of descendants, the depth of the tree, and the number of branches. The model yields the uniform recursive trees as a certain limit, some new results for the uniform case are obtained as well.