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Scaling Limits of Multitype Bienaymé Trees

Authors Louigi Addario-Berry , Philipp Beltran , Benedikt Stufler , Paul Thévenin

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Trees
  • Mathematics of computing → Discrete mathematics
  • Mathematics of computing → Distribution functions
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Combinatorics
Keywords
  • branching processes
  • multitype trees
  • scaling limit

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Abstract

We first consider irreducible critical multitype Bienaymé trees and extend the results to the case, when they possess a critical irreducible component with attached subcritical components. We study these trees under two distinct conditioning frameworks: first, conditioning on the value of a linear combination of the numbers of vertices of given types; and second, conditioning on the precise number of vertices belonging to a selected subset of types. We prove that, under a finite exponential moment condition, the scaling limit as the tree size tends to infinity is given by the Brownian Continuum Random Tree. Additionally, we establish strong non-asymptotic tail bounds for the height of such trees. Our main tools include a flattening operation applied to multitype trees and sharp estimates regarding the structure of monotype trees with a given sequence of degrees.

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Louigi Addario-Berry, Philipp Beltran, Benedikt Stufler, and Paul Thévenin. Scaling Limits of Multitype Bienaymé Trees. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.4

Author Details

Louigi Addario-Berry
  • McGill University, Canada
Philipp Beltran
  • TU Wien, Austria
Benedikt Stufler
  • TU Wien, Austria
Paul Thévenin
  • University of Angers, France

Funding

This research was funded in part by the Austrian Science Fund (FWF) projects 10.55776/ F1002 and 10.55776/PAT6732623 and MSCA-RISE-2020 project RandNET, Contract 101007705. LAB acknowledges support of NSERC and the Canada Research Chairs program. The computational results have been achieved using the Austrian Scientific Computing (ASC) infrastructure. For open access purposes, the authors have applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.

References

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