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題名 多種系統發生網路類別的計數與機率研究
An Enumerative and Probabilistic Study of Various Phylogenetic Networks Classes
作者 張裕昇
Chang, Yu-Sheng
貢獻者 符麥克
Fuchs, Michael
張裕昇
Chang, Yu-Sheng
關鍵詞 系統發生網路
枚舉
漸進計數
組件圖方法
生成函數
奇異點分析
拉普拉斯方法
Phylogenetic network
Enumeration
Asymptotic counting
Component graph method
Generating function
Singularity analysis
Laplace method
日期 2026
上傳時間 2-Mar-2026 13:38:44 (UTC+8)
摘要 本論文題為「An Enumerative and Probabilistic Study of Various Phylogenetic Network Classes」 （多種系統發生網路類別的計數與機率研究）。系統發生網絡(Phylogenetic networks) 是生 物學中使用的圖形模型，用於表示物種或基因之間複雜的演化關係。與傳統的系統發生 樹(Phylogenetic trees) 不同，系統發生網絡能夠捕捉雜交、水平基因轉移、基因組融合和重組 等事件，因此在研究病毒、細菌和植物等複雜演化過程的生物體中非常有用。本研究將這些 問題抽象化為圖論領域的數學問題。 論文主要關注四個核心網絡類別：Tree-child (TC) 網絡、Galled (GN) 網絡、Reticulationvisible (RV) 網絡以及Galled Tree-child (GTC) 網絡，同時也探討了TC網絡的d-結合（dcombining）推廣。 研究所採用的主要方法學包括圖論的基礎概念、用於枚舉分析的組件圖方法 (Component Graph Method, CGM），以及多種漸近方法，例如生成函數、奇異點分析、拉普拉斯方法和 拉格朗日反演公式。 本論文提供了對這些網絡類別的枚舉和性質的深入分析，內容涵蓋以下關鍵主題：確定 網點(reticulations) 數量k 的最緊緻上界（sharp bound）、推導少量k 的封閉形式(closed form)、 計算最大k 的網絡數量、針對固定k 的漸近計數，以及探討這些網絡的機率特性，包括隨機 網絡中k 的極限分佈及Sackin Index 等形狀參數。本研究期望透過這些結果，能加深對系統發 生網絡結構和計數行為的理解。
This thesis, entitled "An Enumerative and Probabilistic Study of Various Phylogenetic Network Classes", is dedicated to advancing the mathematical understanding of phylogenetic networks. Phylogenetic networks are graphical models used in biology to represent complex evolutionary relationships among species or genes. Unlike traditional phylogenetic trees, networks are able to capture events such as hybridization, horizontal gene transfer, genome fusion, and recombination, making them useful in studying organisms like viruses, bacteria, and plants where such processes occur. This work abstracts these types of problems into the realm of graph theory. The study primarily focuses on four main classes of phylogenetic networks: Tree-child (TC) networks, Galled (GN) networks, Reticulation-visible (RV) networks, and Galled Tree-child (GTC) networks, along with a generalization of tree-child networks to the d-combining case. The methodologies employed rely on fundamental concepts from Graph Theory, the crucial graph-theoretic tool for enumeration, the Component Graph Method (CGM), and several Asymptotic Methods, including Generating Functions, Singularity Analysis, the Laplace Method, and the Lagrange Inversion Formula. The research provides in-depth analyses across several key topics: determining sharp bounds for the maximal number of reticulations, deriving closed-form expressions for networks with a small number of reticulations k, counting maximally reticulated networks, conducting asymptotic counting for fixed k and for the total number of networks, and investigating probabilistic properties of these structures, such as the limit distribution of the number of reticulation nodes and the analysis of the Sackin Index. These results enhance the enumerative and probabilistic understanding of phylogenetic network classes.
參考文獻 [1] Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Asymptotic enumeration of rooted binary unlabeled galled trees with a fixed number of galls. In Proceedings of the 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024), pages 27–1. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2024. [2] Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Enumerative combinatorics of unlabeled and labeled time-consistent galled trees. arXiv preprint arXiv:2504.16302, 2025. [3] Lily Agranat-Tamir, Karthik V. Seetharaman, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Combinatorial comparison of general galled trees, time-consistent galled trees, and simplex time-consistent galled trees. Submitted, 2025. [4] Cyril Banderier, Philippe Marchal, and Michael Wallner. Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version). In Lucia Ferrari and Maro Vamvakari, editors, Proceedings of the GASCom 2018 Workshop, volume 2113 of CEUR Workshop Proceedings, pages 60–68, 2018. Also available on arXiv:1805.09017. [5] Cyril Banderier and Michael Wallner. Young tableaux with periodic walls: Counting with the density method. Sém. Lothar. Combin. B, 85, 2021. [6] Edward A. Bender and L. Bruce Richmond. An asymptotic expansion for the coefficients of some power series II: Lagrange inversion. Discrete Mathematics, 50:135–141, 1984. [7] Magnus Bordewich and Charles Semple. Reticulation-visible networks. Advances in Applied Mathematics, 78:114–141, 2016. [8] Gabriel Cardona, Francesc Rosselló, and Gabriel Valiente. Comparison of tree-child phylogenetic networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 6(4):552–569, 2009. [9] Gabriel Cardona and Louxin Zhang. Counting and enumerating tree-child networks and their subclasses. Journal of Computer and System Sciences, 114:84–104, 2020. [10] Yu-Sheng Chang and Michael Fuchs. Counting phylogenetic networks with few reticulation vertices: Galled and reticulation-visible networks. Bulletin of Mathematical Biology, 2024. Originally appeared as a preprint on April 25, 2024; to appear in Bull. Math. Biol. [11] Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu. Enumeration of d-combining tree-child networks. In Proceedings of the 33rd International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022) , volume 225 of LIPIcs. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2022. Paper No. 5. [12] Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu. Enumerative and distributional results for d-combining tree-child networks. Advances in Applied Mathematics, 157:102704, 2024. Originally appeared as a preprint on March 25, 2024. doi:10.1016/j. aam.2024.102704. [13] Yu-Sheng Chang, Michael Fuchs, and Guan-Ru Yu. Galled tree-child networks. In Cécile Mailler and Sebastian Wild, editors, Proceedings of the 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024), volume 2 of Leibniz International Proceedings in Informatics (LIPIcs), pages 2:1–2:13. Schloss Dagstuhl– Leibniz-Zentrum für Informatik, 2024. doi:10.4230/LIPIcs.AofA.2024.8. [14] Andrew Elvey Price, Wenjie Fang, and Michael Wallner. Asymptotics of minimal deterministic finite automata recognizing a finite binary language. In Proceedings of the 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020), pages 11–1. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2020. [15] Andrew Elvey Price, Wenjie Fang, and Michael Wallner. Compacted binary trees admit a stretched exponential. Journal of Combinatorial Theory, Series A, 177:105306, 2021. [16] Mareike Fischer, Lina Herbst, Sophie Kersting, Annemarie Luise Kühn, and Kristina Wicke. Tree balance indices: A comprehensive survey . Springer Cham, 2023. doi:10.1007/978-3-031-39800-1. [17] Mareike Fischer, Lina Herbst, Sophie Kersting, Luise Kühn, and Kristina Wicke. Tree balance indices: A comprehensive survey. arXiv preprint arXiv:2109.12281, 2021. [18] Philippe Flajolet and Robert Sedgewick. Analytic combinatorics. Cambridge University Press,2009. [19] Michael Fuchs and Bernhard Gittenberger. Sackin indices for labeled and unlabeled classes of galled trees. Journal of Mathematical Biology, 90(4):1–34, 2025. [20] Michael Fuchs, Bernhard Gittenberger, and Marefatollah Mansouri. Counting phylogenetic networks with few reticulation vertices: Tree-child and normal networks. arXiv preprint arXiv:1803.11325, 2018. [21] Michael Fuchs, Bernhard Gittenberger, and Marefatollah Mansouri. Counting phylogenetic networks with few reticulation vertices: Exact enumeration and corrections. arXiv preprint arXiv:2006.15784, 2020. [22] Michael Fuchs, En-Yu Huang, and Guan-Ru Yu. Counting phylogenetic networks with few reticulation vertices: A second approach. Discrete Applied Mathematics, 320:140–149, 2022. [23] Michael Fuchs, Hexuan Liu, and Guan-Ru Yu. A short note on the exact counting of tree-child networks. arXiv preprint arXiv:2110.03842, 2021. [24] Michael Fuchs, Guan-Ru Yu, and Louxin Zhang. On the asymptotic growth of the number of tree-child networks. European Journal of Combinatorics, 93:103278, 2021. [25] Michael Fuchs, Guan-Ru Yu, and Louxin Zhang. Asymptotic enumeration and distributional properties of galled networks. Journal of Combinatorial Theory, Series A, 189:105599, 2022. [26] Philippe Gambette, Andreas D. M. Gunawan, Anthony Labarre, Stéphane Vialette, and Louxin Zhang. Locating a tree in a phylogenetic network in quadratic time. In Research in Computational Molecular Biology: 19th Annual International Conference, RECOMB 2015, Proceedings, volume 9029 of Lecture Notes in Computer Science, pages 96–107. Springer, 2015. doi:10.1007/978-3-319-16706-0_9. [27] Philippe Gambette, Andreas D. M. Gunawan, Anthony Labarre, Stéphane Vialette, and Louxin Zhang. Solving the tree containment problem in linear time for nearly stable phylogenetic networks. Discrete Applied Mathematics, 246:62–79, 2018. [28] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete mathematics: A foundation for computer science. Addison-Wesley, 2nd edition, 1994. [29] Andreas D. M. Gunawan, Bhaskar DasGupta, and Louxin Zhang. A decomposition theorem and two algorithms for reticulation-visible networks. Information and Computation, 252:161–175, 2017. [30] Andreas D. M. Gunawan, Jeyaram Rathin, and Louxin Zhang. Counting and enumerating galled networks. Discrete Applied Mathematics, 283:644–654, 2020. [31] Andreas D. M. Gunawan, Hongwei Yan, and Louxin Zhang. Compression of phylogenetic networks and algorithm for the tree containment problem. Journal of Computational Biology, 26(3):285–294, 2019. [32] Andreas D. M. Gunawan and Louxin Zhang. Bounding the size of a network defined by visibility property. arXiv preprint arXiv:1510.00115, 2015. [33] Daniel H. Huson and Tobias H. Klöpper. Beyond galled trees-decomposition and computation of galled networks. In Annual International Conference on Research in Computational Molecular Biology, pages 211–225. Springer, 2007. [34] Sungsik Kong, Joan Carles Pons, Laura Kubatko, and Kristina Wicke. Classes of explicit phylogenetic networks and their biological and mathematical significance. Journal of Mathematical Biology, 84(6):47, 2022. [35] Marefatollah Mansouri. Counting general phylogenetic networks. arXiv preprint arXiv:2005.14547, 2020. [36] Colin McDiarmid, Charles Semple, and Dominic Welsh. Counting phylogenetic networks. Annals of Combinatorics, 19(1):205–224, 2015. [37] Miquel Pons and Josep Batle. Combinatorial characterization of a certain class of words and a conjectured connection with general subclasses of phylogenetic tree-child networks. Scientific Reports, 11(1):21875, 2021. [38] Charles Semple. Size of a phylogenetic network. Discrete Applied Mathematics, 217:362–367, 2017. [39] Mike Steel. Phylogeny: Discrete and random processes in evolution. SIAM, 2016. [40] Erlang Surya and Lutz Warnke. Lagrange inversion formula by induction. The American Mathematical Monthly, 130:1–5, 09 2023. doi:10.1080/00029890.2023.2251344. [41] Michael Wallner. Ancillary files of arXiv:2203.07619. http://dmg.tuwien.ac.at/ mwallner, 2022. DOI: 10.48550/arXiv.2203.07619. [42] Stephen J. Willson. Properties of normal phylogenetic networks. Bulletin of Mathematical Biology, 72(2):340–358, 2010. [43] Louxin Zhang. The Sackin index of simplex networks. In RECOMB International Workshop on Comparative Genomics, pages 52–67. Springer, 2022.
描述 博士
國立政治大學
應用數學系
109751502
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109751502
資料類型 thesis
dc.contributor.advisor 符麥克zh_TW
dc.contributor.advisor Fuchs, Michaelen_US
dc.contributor.author (Authors) 張裕昇zh_TW
dc.contributor.author (Authors) Chang, Yu-Shengen_US
dc.creator (作者) 張裕昇zh_TW
dc.creator (作者) Chang, Yu-Shengen_US
dc.date (日期) 2026en_US
dc.date.accessioned 2-Mar-2026 13:38:44 (UTC+8)-
dc.date.available 2-Mar-2026 13:38:44 (UTC+8)-
dc.date.issued (上傳時間) 2-Mar-2026 13:38:44 (UTC+8)-
dc.identifier (Other Identifiers) G0109751502en_US
dc.identifier.uri (URI) https://nccur.lib.nccu.edu.tw/handle/140.119/161932-
dc.description (描述) 博士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學系zh_TW
dc.description (描述) 109751502zh_TW
dc.description.abstract (摘要) 本論文題為「An Enumerative and Probabilistic Study of Various Phylogenetic Network Classes」 （多種系統發生網路類別的計數與機率研究）。系統發生網絡(Phylogenetic networks) 是生 物學中使用的圖形模型，用於表示物種或基因之間複雜的演化關係。與傳統的系統發生 樹(Phylogenetic trees) 不同，系統發生網絡能夠捕捉雜交、水平基因轉移、基因組融合和重組 等事件，因此在研究病毒、細菌和植物等複雜演化過程的生物體中非常有用。本研究將這些 問題抽象化為圖論領域的數學問題。 論文主要關注四個核心網絡類別：Tree-child (TC) 網絡、Galled (GN) 網絡、Reticulationvisible (RV) 網絡以及Galled Tree-child (GTC) 網絡，同時也探討了TC網絡的d-結合（dcombining）推廣。 研究所採用的主要方法學包括圖論的基礎概念、用於枚舉分析的組件圖方法 (Component Graph Method, CGM），以及多種漸近方法，例如生成函數、奇異點分析、拉普拉斯方法和 拉格朗日反演公式。 本論文提供了對這些網絡類別的枚舉和性質的深入分析，內容涵蓋以下關鍵主題：確定 網點(reticulations) 數量k 的最緊緻上界（sharp bound）、推導少量k 的封閉形式(closed form)、 計算最大k 的網絡數量、針對固定k 的漸近計數，以及探討這些網絡的機率特性，包括隨機 網絡中k 的極限分佈及Sackin Index 等形狀參數。本研究期望透過這些結果，能加深對系統發 生網絡結構和計數行為的理解。zh_TW
dc.description.abstract (摘要) This thesis, entitled "An Enumerative and Probabilistic Study of Various Phylogenetic Network Classes", is dedicated to advancing the mathematical understanding of phylogenetic networks. Phylogenetic networks are graphical models used in biology to represent complex evolutionary relationships among species or genes. Unlike traditional phylogenetic trees, networks are able to capture events such as hybridization, horizontal gene transfer, genome fusion, and recombination, making them useful in studying organisms like viruses, bacteria, and plants where such processes occur. This work abstracts these types of problems into the realm of graph theory. The study primarily focuses on four main classes of phylogenetic networks: Tree-child (TC) networks, Galled (GN) networks, Reticulation-visible (RV) networks, and Galled Tree-child (GTC) networks, along with a generalization of tree-child networks to the d-combining case. The methodologies employed rely on fundamental concepts from Graph Theory, the crucial graph-theoretic tool for enumeration, the Component Graph Method (CGM), and several Asymptotic Methods, including Generating Functions, Singularity Analysis, the Laplace Method, and the Lagrange Inversion Formula. The research provides in-depth analyses across several key topics: determining sharp bounds for the maximal number of reticulations, deriving closed-form expressions for networks with a small number of reticulations k, counting maximally reticulated networks, conducting asymptotic counting for fixed k and for the total number of networks, and investigating probabilistic properties of these structures, such as the limit distribution of the number of reticulation nodes and the analysis of the Sackin Index. These results enhance the enumerative and probabilistic understanding of phylogenetic network classes.en_US
dc.description.tableofcontents 1 Introduction 6 1.1 Graph Theory 6 1.1.1 Basic Knowledge 6 1.1.2 Classes of Networks 11 1.1.3 The Component Graph Method (CGM) 18 1.1.3.1 d-combining tree-child networks 19 1.1.3.2 Galled networks 23 1.1.3.3 Reticulation-visible networks 28 1.1.3.4 Galled tree-child networks 32 1.1.4 Dup-trees 35 1.2 Asymptotic Methods 40 1.2.1 Generating Functions 40 1.2.2 Singularity Analysis 44 1.2.3 Laplace Method 53 1.2.4 Lagrange Inversion Formula 57 2 Old and New Results for the Main Classes 61 2.1 The maximal number of reticulations 62 2.1.1 Reticulation-visible networks 66 2.1.2 Galled tree-child networks 69 2.2 Closed-form expressions for small k and arbitrary n 70 2.2.1 d-combining tree-child networks 70 2.2.2 Galled networks 74 2.2.3 Reticulation-visible networks 78 2.2.4 Galled tree-child networks 82 2.3 Exact counting for any k and any n 84 2.3.1 One-component d-combining tree-child networks 84 2.3.2 d-combining tree-child networks 86 2.4 Counting maximally reticulated networks 94 2.4.1 d-combining tree-child networks 94 2.4.2 Galled tree-child networks 95 2.5 Asymptotic counting for fixed k as n → ∞ 97 2.5.1 d-combing tree-child networks 97 2.5.2 Reticulation-visible networks 99 2.5.3 Galled networks 100 2.5.4 Galled tree-child networks 101 2.6 Asymptotic counting of the total number of networks 102 2.6.1 One-component d-combining tree-child networks 103 2.6.2 d-combining tree-child networks 103 2.6.3 Galled tree-child networks 109 2.7 The number of reticulations for a random network 112 2.7.1 One-component d-combining tree-child network 112 2.7.2 d-combing tree-child networks 114 2.7.2.1 Bi-combining networks 115 2.7.2.2 d-combining networks with d ≥ 3 119 2.7.3 Galled tree-child networks 124 2.8 Sackin Index 125 2.8.1 One-component d-combining tree-child networks 125 2.9 Other 136 2.9.1 Corollaries of Theorem 60 (d-combining tree-child network) 136 2.9.2 Proofs of Propositions 16 and 17 (d-combining tree-child network) 137 3 Conclusion and Outlook 140 Bibliography 143 Appendix A Trip-trees 148zh_TW
dc.format.extent 1912527 bytes-
dc.format.mimetype application/pdf-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109751502en_US
dc.subject (關鍵詞) 系統發生網路zh_TW
dc.subject (關鍵詞) 枚舉zh_TW
dc.subject (關鍵詞) 漸進計數zh_TW
dc.subject (關鍵詞) 組件圖方法zh_TW
dc.subject (關鍵詞) 生成函數zh_TW
dc.subject (關鍵詞) 奇異點分析zh_TW
dc.subject (關鍵詞) 拉普拉斯方法zh_TW
dc.subject (關鍵詞) Phylogenetic networken_US
dc.subject (關鍵詞) Enumerationen_US
dc.subject (關鍵詞) Asymptotic countingen_US
dc.subject (關鍵詞) Component graph methoden_US
dc.subject (關鍵詞) Generating functionen_US
dc.subject (關鍵詞) Singularity analysisen_US
dc.subject (關鍵詞) Laplace methoden_US
dc.title (題名) 多種系統發生網路類別的計數與機率研究zh_TW
dc.title (題名) An Enumerative and Probabilistic Study of Various Phylogenetic Networks Classesen_US
dc.type (資料類型) thesisen_US
dc.relation.reference (參考文獻) [1] Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Asymptotic enumeration of rooted binary unlabeled galled trees with a fixed number of galls. In Proceedings of the 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024), pages 27–1. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2024. [2] Lily Agranat-Tamir, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Enumerative combinatorics of unlabeled and labeled time-consistent galled trees. arXiv preprint arXiv:2504.16302, 2025. [3] Lily Agranat-Tamir, Karthik V. Seetharaman, Michael Fuchs, Bernhard Gittenberger, and Noah A. Rosenberg. Combinatorial comparison of general galled trees, time-consistent galled trees, and simplex time-consistent galled trees. Submitted, 2025. [4] Cyril Banderier, Philippe Marchal, and Michael Wallner. Rectangular Young tableaux with local decreases and the density method for uniform random generation (short version). In Lucia Ferrari and Maro Vamvakari, editors, Proceedings of the GASCom 2018 Workshop, volume 2113 of CEUR Workshop Proceedings, pages 60–68, 2018. Also available on arXiv:1805.09017. [5] Cyril Banderier and Michael Wallner. Young tableaux with periodic walls: Counting with the density method. Sém. Lothar. Combin. B, 85, 2021. [6] Edward A. Bender and L. Bruce Richmond. An asymptotic expansion for the coefficients of some power series II: Lagrange inversion. Discrete Mathematics, 50:135–141, 1984. [7] Magnus Bordewich and Charles Semple. Reticulation-visible networks. Advances in Applied Mathematics, 78:114–141, 2016. [8] Gabriel Cardona, Francesc Rosselló, and Gabriel Valiente. Comparison of tree-child phylogenetic networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 6(4):552–569, 2009. [9] Gabriel Cardona and Louxin Zhang. Counting and enumerating tree-child networks and their subclasses. Journal of Computer and System Sciences, 114:84–104, 2020. [10] Yu-Sheng Chang and Michael Fuchs. Counting phylogenetic networks with few reticulation vertices: Galled and reticulation-visible networks. Bulletin of Mathematical Biology, 2024. Originally appeared as a preprint on April 25, 2024; to appear in Bull. Math. Biol. [11] Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu. Enumeration of d-combining tree-child networks. In Proceedings of the 33rd International Meeting on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2022) , volume 225 of LIPIcs. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2022. Paper No. 5. [12] Yu-Sheng Chang, Michael Fuchs, Hexuan Liu, Michael Wallner, and Guan-Ru Yu. Enumerative and distributional results for d-combining tree-child networks. Advances in Applied Mathematics, 157:102704, 2024. Originally appeared as a preprint on March 25, 2024. doi:10.1016/j. aam.2024.102704. [13] Yu-Sheng Chang, Michael Fuchs, and Guan-Ru Yu. Galled tree-child networks. In Cécile Mailler and Sebastian Wild, editors, Proceedings of the 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024), volume 2 of Leibniz International Proceedings in Informatics (LIPIcs), pages 2:1–2:13. Schloss Dagstuhl– Leibniz-Zentrum für Informatik, 2024. doi:10.4230/LIPIcs.AofA.2024.8. [14] Andrew Elvey Price, Wenjie Fang, and Michael Wallner. Asymptotics of minimal deterministic finite automata recognizing a finite binary language. In Proceedings of the 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020), pages 11–1. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2020. [15] Andrew Elvey Price, Wenjie Fang, and Michael Wallner. 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