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**Published in:** LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Composition schemes are ubiquitous in combinatorics, statistical mechanics and probability theory. We give a unifying explanation to various phenomena observed in the combinatorial and statistical physics literature in the context of q-enumeration (this is a model where objects with a parameter of value k have a Gibbs measure/Boltzmann weight q^k). For structures enumerated by a composition scheme, we prove a phase transition for any parameter having such a Gibbs measure: for a critical value q = q_c, the limit law of the parameter is a two-parameter Mittag-Leffler distribution, while it is Gaussian in the supercritical regime (q > q_c), and it is a Boltzmann distribution in the subcritical regime (0 < q < q_c). We apply our results to fundamental statistics of lattice paths and quarter-plane walks. We also explain previously observed limit laws for pattern-restricted permutations, and a phenomenon uncovered by Krattenthaler for the wall contacts in watermelons.

Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner. Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{banderier_et_al:LIPIcs.AofA.2024.7, author = {Banderier, Cyril and Kuba, Markus and Wagner, Stephan and Wallner, Michael}, title = {{Composition Schemes: q-Enumerations and Phase Transitions in Gibbs Models}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {7:1--7:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-329-4}, ISSN = {1868-8969}, year = {2024}, volume = {302}, editor = {Mailler, C\'{e}cile and Wild, Sebastian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.7}, URN = {urn:nbn:de:0030-drops-204427}, doi = {10.4230/LIPIcs.AofA.2024.7}, annote = {Keywords: Composition schemes, q-enumeration, generating functions, Gibbs distribution, phase transitions, limit laws, Mittag-Leffler distribution, chi distribution, Boltzmann distribution} }

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**Published in:** LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

A B-tree is a type of search tree where every node (except possibly for the root) contains between m and 2m keys for some positive integer m, and all leaves have the same distance to the root. We study sequences of B-trees that can arise from successively inserting keys, and in particular present a bijection between such sequences (which we call histories) and a special type of increasing trees. We describe the set of permutations for the keys that belong to a given history, and also show how to use this bijection to analyse statistics associated with B-trees.

Fabian Burghart and Stephan Wagner. A Bijection for the Evolution of B-Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{burghart_et_al:LIPIcs.AofA.2024.10, author = {Burghart, Fabian and Wagner, Stephan}, title = {{A Bijection for the Evolution of B-Trees}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-329-4}, ISSN = {1868-8969}, year = {2024}, volume = {302}, editor = {Mailler, C\'{e}cile and Wild, Sebastian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.10}, URN = {urn:nbn:de:0030-drops-204451}, doi = {10.4230/LIPIcs.AofA.2024.10}, annote = {Keywords: B-trees, histories, increasing trees, bijection, asymptotic enumeration, tree statistics} }

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**Published in:** LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

A fringe subtree of a rooted tree is a subtree that consists of a vertex and all its descendants. The number of distinct fringe subtrees in random trees has been studied by several authors, notably because of its connection to tree compaction algorithms. Here, we obtain a very precise result for binary search trees: it is shown that the number of distinct fringe subtrees in a binary search tree with n leaves is asymptotically equal to (c₁n)/(log n) for a constant c₁ ≈ 2.4071298335, both in expectation and with high probability. This was previously shown to be a lower bound, our main contribution is to prove a matching upper bound. The method is quite general and can also be applied to similar problems for other tree models.

Stephan Wagner. On the Number of Distinct Fringe Subtrees in Binary Search Trees. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 13:1-13:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{wagner:LIPIcs.AofA.2024.13, author = {Wagner, Stephan}, title = {{On the Number of Distinct Fringe Subtrees in Binary Search Trees}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {13:1--13:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-329-4}, ISSN = {1868-8969}, year = {2024}, volume = {302}, editor = {Mailler, C\'{e}cile and Wild, Sebastian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.13}, URN = {urn:nbn:de:0030-drops-204482}, doi = {10.4230/LIPIcs.AofA.2024.13}, annote = {Keywords: Fringe subtrees, binary search trees, tree compression, minimal DAG, asymptotics} }

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**Published in:** LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

Making use of a newly developed package in the computer algebra system SageMath, we show how to perform a full asymptotic analysis by means of the Mellin transform with explicit error bounds. As an application of the method, we answer a question of Bóna and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum.

Benjamin Hackl and Stephan Wagner. Binomial Sums and Mellin Asymptotics with Explicit Error Bounds: A Case Study. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hackl_et_al:LIPIcs.AofA.2024.19, author = {Hackl, Benjamin and Wagner, Stephan}, title = {{Binomial Sums and Mellin Asymptotics with Explicit Error Bounds: A Case Study}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {19:1--19:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-329-4}, ISSN = {1868-8969}, year = {2024}, volume = {302}, editor = {Mailler, C\'{e}cile and Wild, Sebastian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.19}, URN = {urn:nbn:de:0030-drops-204549}, doi = {10.4230/LIPIcs.AofA.2024.19}, annote = {Keywords: binomial sum, Mellin transform, asymptotics, explicit error bounds, B-terms} }

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**Published in:** LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

In this paper we obtain some new results on the enumeration of parking functions and labeled forests. We introduce new statistics both for parking functions and for labeled forests that are connected to each other by means of a bijection. We determine the joint distribution of two statistics on parking functions and their counterparts on labeled forests. Our results on labeled forests also serve to explain the mysterious equidistribution between two seemingly unrelated statistics in parking functions recently identified by Stanley and Yin and give an explicit bijection between the two statistics.

Stephan Wagner and Mei Yin. Statistics of Parking Functions and Labeled Forests. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{wagner_et_al:LIPIcs.AofA.2024.29, author = {Wagner, Stephan and Yin, Mei}, title = {{Statistics of Parking Functions and Labeled Forests}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-329-4}, ISSN = {1868-8969}, year = {2024}, volume = {302}, editor = {Mailler, C\'{e}cile and Wild, Sebastian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.29}, URN = {urn:nbn:de:0030-drops-204648}, doi = {10.4230/LIPIcs.AofA.2024.29}, annote = {Keywords: parking function, labeled forest, generating function, Pollak’s circle argument, bijection} }

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**Published in:** LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

We consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with n vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this extended abstract: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling. Second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase. Lastly, the largest connected component, for which we also observe a phase transition.

Benjamin Hackl, Alois Panholzer, and Stephan Wagner. Uncovering a Random Tree. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 10:1-10:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hackl_et_al:LIPIcs.AofA.2022.10, author = {Hackl, Benjamin and Panholzer, Alois and Wagner, Stephan}, title = {{Uncovering a Random Tree}}, booktitle = {33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)}, pages = {10:1--10:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-230-3}, ISSN = {1868-8969}, year = {2022}, volume = {225}, editor = {Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.10}, URN = {urn:nbn:de:0030-drops-160962}, doi = {10.4230/LIPIcs.AofA.2022.10}, annote = {Keywords: Labeled tree, uncover process, functional central limit theorem, limiting distribution, phase transition} }

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**Published in:** LIPIcs, Volume 225, 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)

We study the size of the automorphism group of two different types of random trees: Galton-Watson trees and Pólya trees. In both cases, we prove that it asymptotically follows a log-normal distribution. While the proof for Galton-Watson trees mainly relies on probabilistic arguments and a general result on additive tree functionals, generating functions are used in the case of Pólya trees.

Christoffer Olsson and Stephan Wagner. Automorphisms of Random Trees. In 33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 225, pp. 16:1-16:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{olsson_et_al:LIPIcs.AofA.2022.16, author = {Olsson, Christoffer and Wagner, Stephan}, title = {{Automorphisms of Random Trees}}, booktitle = {33rd International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-230-3}, ISSN = {1868-8969}, year = {2022}, volume = {225}, editor = {Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2022.16}, URN = {urn:nbn:de:0030-drops-161026}, doi = {10.4230/LIPIcs.AofA.2022.16}, annote = {Keywords: random tree, Galton-Watson tree, P\'{o}lya tree, automorphism group, central limit theorem} }

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**Published in:** LIPIcs, Volume 159, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)

We study block statistics in subcritical graph classes; these are statistics that can be defined as the sum of a certain weight function over all blocks. Examples include the number of edges, the number of blocks, and the logarithm of the number of spanning trees. The main result of this paper is a central limit theorem for statistics of this kind under fairly mild technical assumptions.

Dimbinaina Ralaivaosaona, Clément Requilé, and Stephan Wagner. Block Statistics in Subcritical Graph Classes. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ralaivaosaona_et_al:LIPIcs.AofA.2020.24, author = {Ralaivaosaona, Dimbinaina and Requil\'{e}, Cl\'{e}ment and Wagner, Stephan}, title = {{Block Statistics in Subcritical Graph Classes}}, booktitle = {31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-147-4}, ISSN = {1868-8969}, year = {2020}, volume = {159}, editor = {Drmota, Michael and Heuberger, Clemens}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2020.24}, URN = {urn:nbn:de:0030-drops-120543}, doi = {10.4230/LIPIcs.AofA.2020.24}, annote = {Keywords: subcritical graph class, block statistic, number of blocks, number of edges, number of spanning trees} }

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**Published in:** LIPIcs, Volume 159, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)

Given a positive integer n and a real number p ∈ [0,1], let D(n,p) denote the random digraph defined in the following way: each of the binom(n,2) possible edges on the vertex set {1,2,3,…,n} is included with probability 2p, where all edges are independent of each other. Thereafter, a direction is chosen independently for each edge, with probability 1/2 for each possible direction. In this paper, we study the probability that a random instance of D(n,p) is acyclic, i.e., that it does not contain a directed cycle. We find precise asymptotic formulas for the probability of a random digraph being acyclic in the sparse regime, i.e., when np = O(1). As an example, for each real number μ, we find an exact analytic expression for φ(μ) = lim_{n→ ∞} n^{1/3} ℙ{D(n,1/n (1+μ n^{-1/3})) is acyclic}.

Dimbinaina Ralaivaosaona, Vonjy Rasendrahasina, and Stephan Wagner. On the Probability That a Random Digraph Is Acyclic. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 25:1-25:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ralaivaosaona_et_al:LIPIcs.AofA.2020.25, author = {Ralaivaosaona, Dimbinaina and Rasendrahasina, Vonjy and Wagner, Stephan}, title = {{On the Probability That a Random Digraph Is Acyclic}}, booktitle = {31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)}, pages = {25:1--25:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-147-4}, ISSN = {1868-8969}, year = {2020}, volume = {159}, editor = {Drmota, Michael and Heuberger, Clemens}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2020.25}, URN = {urn:nbn:de:0030-drops-120557}, doi = {10.4230/LIPIcs.AofA.2020.25}, annote = {Keywords: Random digraphs, acyclic digraphs, asymptotics} }

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**Published in:** LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

Tanglegrams are structures consisting of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the two trees. We say that a tanglegram is planar if it can be drawn in the plane without crossings. Using a blend of combinatorial and analytic techniques, we determine an asymptotic formula for the number of planar tanglegrams with n leaves on each side.

Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana, and Stephan Wagner. Counting Planar Tanglegrams. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ralaivaosaona_et_al:LIPIcs.AofA.2018.32, author = {Ralaivaosaona, Dimbinaina and Ravelomanana, Jean Bernoulli and Wagner, Stephan}, title = {{Counting Planar Tanglegrams}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {32:1--32:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-078-1}, ISSN = {1868-8969}, year = {2018}, volume = {110}, editor = {Fill, James Allen and Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.32}, URN = {urn:nbn:de:0030-drops-89259}, doi = {10.4230/LIPIcs.AofA.2018.32}, annote = {Keywords: rooted binary trees, tanglegram, planar, generating functions, asymptotic enumeration, singularity analysis} }

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**Published in:** LIPIcs, Volume 110, 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)

An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are almost local, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger.

Dimbinaina Ralaivaosaona, Matas Sileikis, and Stephan Wagner. Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ralaivaosaona_et_al:LIPIcs.AofA.2018.33, author = {Ralaivaosaona, Dimbinaina and Sileikis, Matas and Wagner, Stephan}, title = {{Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees}}, booktitle = {29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)}, pages = {33:1--33:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-078-1}, ISSN = {1868-8969}, year = {2018}, volume = {110}, editor = {Fill, James Allen and Ward, Mark Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2018.33}, URN = {urn:nbn:de:0030-drops-89262}, doi = {10.4230/LIPIcs.AofA.2018.33}, annote = {Keywords: Galton-Watson trees, central limit theorem, additive functional} }

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