Cost functionals for large (uniform and simply generated) random trees

TL;DR

This paper establishes an invariance principle for additive tree functionals in large random trees, including Catalan and simply generated models, connecting combinatorial structures with probabilistic limits like Brownian excursions and stable Lévy trees.

Contribution

It provides a unified invariance principle for additive functionals in both Catalan and simply generated trees, extending previous results to the stable case.

Findings

Invariance principle for Catalan trees using Brownian excursion embedding

Extension of results to simply generated trees via convergence to stable Lévy trees

Generalization of quadratic case results to stable cases in tree functionals

Abstract

Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of internal nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary second moment computations. We recover results first given by Fill and Kapur (2004) and then by Fill and Janson (2009). In the simply generated case, this relies on the convergence of conditioned Galton-Watson towards stable L\'evy trees. We recover results first given by Janson (2003 and 2016) in the quadratic case and give a generalization to the stable case.