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DOI: 10.4230/LIPIcs.AofA.2026.4

Scaling Limits of Multitype Bienaymé Trees

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

Published in: LIPIcs, Volume 381, 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)

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)

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@InProceedings{addarioberry_et_al:LIPIcs.AofA.2026.4,
  author =	{Addario-Berry, Louigi and Beltran, Philipp and Stufler, Benedikt and Th\'{e}venin, Paul},
  title =	{{Scaling Limits of Multitype Bienaym\'{e} Trees}},
  booktitle =	{37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
  pages =	{4:1--4:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-435-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{381},
  editor =	{Panagiotou, Konstantinos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2026.4},
  URN =		{urn:nbn:de:0030-drops-262750},
  doi =		{10.4230/LIPIcs.AofA.2026.4},
  annote =	{Keywords: branching processes, multitype trees, scaling limit}
}

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Scaling Limits of Multitype Bienaymé Trees

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