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Asymptotic Analysis of Shape Parameters of Trees and Lattice Paths

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Asymptotic Analysis of Shape Parameters of Trees and Lattice Paths Benjamin Hackl Asymptotic Analysis of Shape Parameters of Trees and Lattice Paths DISSERTATION zur Erlangung des akademischen Grades Doktor der Technischen Wissenschaften STUDIUM Doktoratsstudium der Technischen Wissenschaften im Dissertationsgebiet „Technische Mathematik“ Alpen-Adria-Universität Klagenfurt Fakultät für Technische Wissenschaften BETREUER UND ERSTGUTACHTER ZWEITGUTACHTER Univ.-Prof. Dr. Clemens Heuberger Prof. Dr. Stephan Wagner Alpen-Adria-Universität Klagenfurt Stellenbosch University Institut für Mathematik Department of Mathematical Sciences Klagenfurt, 11. Jänner 2018 iii Eidesstattliche Erklärung Ich versichere an Eides statt, dass ich die eingereichte wissenschaftliche Arbeit selbstständig verfasst und keine anderen als • die angegebenen Hilfsmittel benutzt habe, die während des Arbeitsvorganges von dritter Seite erfahrene Unterstützung, ein- • schließlich signifikanter Betreuungshinweise, vollständig offengelegt habe, die Inhalte, die ich aus Werken Dritter oder eigenen Werken wortwörtlich oder sinnge- • mäß übernommen habe, in geeigneter Form gekennzeichnet und den Ursprung der Information durch möglichst exakte Quellenangaben (z.B. in Fußnoten) ersichtlich gemacht habe, die eingereichte wissenschaftliche Arbeit bisher weder im Inland noch im Ausland • einer Prüfungsbehörde vorgelegt habe und bei der Weitergabe jedes Exemplars (z.B. in gebundener, gedruckter oder digitaler • Form) der wissenschaftlichen Arbeit sicherstelle, dass diese mit der eingereichten digitalen Version übereinstimmt. Mir ist bekannt, dass die digitale Version der eingereichten wissenschaftlichen Arbeit zur Plagiatskontrolle herangezogen wird. Ich bin mir bewusst, dass eine tatsachenwidrige Erklärung rechtliche Folgen haben wird. Benjamin Hackl, e.h. Klagenfurt, 11. Jänner 2018 v Acknowledgements The following lines are devoted to my family, friends, and colleagues, without whose support composing this thesis would not have been possible. In particular, I want to thank my advisor Clemens Heuberger for his continuous support, as well as for all the time and resources he invested in this project. Fortunately, we have a very harmonious workplace environment in our department, which I want to thank my colleagues for. Especially, I would like to thank Daniel Krenn, not only for his valuable feedback on this thesis, but also for the non-mathematical guidance he provided. I am very grateful to Helmut Prodinger for supporting me as a mentor within the Young- Scientists-Mentoring-Program of Alpen-Adria-Universität Klagenfurt, and in addition I want to thank him and Stephan Wagner for giving me possibilities of research stays abroad. Finally, I am very grateful to my family, my friends, and in particular my girlfriend Anja for their continuous support—even (and especially) in times where I could not stop talking about mathematics. Thank you! This work was supported financially by the Austrian Science Fund (FWF): P 24644-N26 and by the Karl Popper Kolleg “Modeling-Simulation-Optimization” funded by Alpen-Adria- Universität Klagenfurt together with the Carinthian Economic Promotion Fund (KWF). Addi- tional funding for research stays abroad has been provided by the Mobilitätsförderung für NachwuchswissenschaftlerInnen of Alpen-Adria-Universität Klagenfurt. vi Abstract This thesis belongs to the mathematical field of Analytic Combinatorics, which is concerned with the asymptotic analysis of parameters of discrete structures (here, primarily trees and lattice paths) using analytic methods. After modeling the parameter of interest as a random variable, the properties brought to light by means of such a rigorous investigation include high-precision asymptotic (sometimes even explicit) formulas for the expected value, the corresponding variance—and, if possible, higher moments and the characterization of a limiting distribution as well. Predominantly, the shape parameters under investigation are associated to deterministic reduction procedures defined on families of plane trees and lattice paths, respectively. To be more precise, a suitable deterministic reduction naturally induces some sort of age on the objects (in the sense that “older” objects require more reductions until they are “irreducible”). A prominent example for a parameter that can be modeled in this way is the well-known register function for binary trees. Both the age itself as well as the object size after a fixed number of reductions are studied in different context within this thesis. Another interesting shape parameter is defined for so-called Łukasiewicz paths, i.e., two- dimensional simple lattice paths with a unique down step. These paths have a very nice structure, as they are strongly related to plane trees whose node degrees are contained in a predefined set. In this fairly general setting we are interested in ascents—maximal sequences of non-negative steps. Something all of our investigations have in common is that they all contain some compre- hensive computational aspects. To this end, we make heavy use of the free open-source computer mathematics software system SageMath and its included module for computations with asymptotic expansions developed by Clemens Heuberger, Daniel Krenn, and the author. For all results obtained with the help of this module, there is a corresponding worksheet containing the computations available for download. vii Zusammenfassung Die vorliegende Dissertation ist dem mathematischen Teilgebiet der analytischen Kombinatorik zuzuordnen, welches sich mit der präzisen asymptotischen Analyse von Parametern diskreter Strukturen (in dieser Arbeit konkret Bäumen und Gitterpfaden) mittels analytischer Methoden beschäftigt. Die Ergebnisse der Untersuchungen dieser als Zufallsvariablen modellierten Parameter umfassen asymptotische Ausdrücke hoher Präzision (sowie gegebenenfalls sogar explizite Ausdrücke) für den Erwartungswert, die Varianz — und, sofern möglich, auch für höhere Momente (was gegebenenfalls auch Rückschlüsse auf eine Grenzverteilung erlaubt). Die in erster Linie untersuchten Parameter hängen mit deterministischen Reduktionsproze- duren zusammen, die auf Familien von geordneten Wurzelbäumen, beziehungsweise auf Familien von Gitterpfaden, definiert werden. Genauer gesagt induziert eine geeignete deter- ministische Reduktion auf natürliche Art und Weise ein Alter auf den jeweiligen Objekten (in dem Sinne, dass „ältere“ Objekte öfter reduziert werden müssen, bis sie „irreduzibel“ sind). Ein prominentes Beispiel für einen Parameter, der als ein solches Alter gesehen werden kann, ist die wohlbekannte Registerfunktion binärer Bäume. Sowohl das Alter als auch die Objektgröße nach einer festen Anzahl von Reduktionen werden im Rahmen dieser Arbeit in verschiedenen Kontexten untersucht. Ein weiterer Parameter von Interesse ist für sogenannte Łukasiewicz-Pfade definiert. Das sind einfache zweidimensionale Gitterpfade, bei denen die Menge der erlaubten Schritte nur einen einzigen Schritt nach unten enthält. Die Struktur von Łukasiewicz-Pfaden ist besonders reichhaltig, weil sie stark mit jenen geordneten Wurzelbäumen, deren Knotengrade in einer vorgegebenen Menge enthalten sind, zusammenhängen. In diesem relativ allgemeinen Rahmen werden Aufstiege — das sind maximale Folgen nicht-negativer Schritte — untersucht. Ein Aspekt, der allen Analysen in dieser Dissertation gleicherweise innewohnt, ist, dass immer wieder rechentechnisch sehr aufwändige Berechnungen vorkommen. Aus diesem Grund wird starker Gebrauch vom freien, quelloffenen Computermathematiksystem SageMath und insbesondere dem darin enthaltenen Modul für asymptotische Entwicklungen, welches von Clemens Heuberger, Daniel Krenn und dem Autor entwickelt wurde, gemacht. Für alle Resultate, die mit Hilfe dieses Moduls erhalten wurden, steht ein SageMath-Worksheet mit den zugehörigen Berechnungen zum Download zur Verfügung. Contents 1 Introduction 1 2 Reductions of Binary Trees and Lattice Paths 7 2.1 Introduction . .7 2.1.1 Binary Trees . .8 2.1.2 Lattice Paths . .9 2.2 Tree Reductions and the Register Function . 10 2.2.1 Motivation and Preliminaries . 10 2.2.2 r-branches . 17 2.2.3 Total Number of Branches . 25 2.3 Reduction of Lattice Paths . 30 2.3.1 Iterative Reductions and an Analogue to the Register Function . 30 2.3.2 Fringes . 40 3 Cutting and Pruning Plane Trees 49 3.1 Introduction . 49 3.2 Cutting Leaves . 52 3.2.1 Preliminaries . 52 3.2.2 Leaf-Reduction and the Expansion Operator . 56 3.2.3 Asymptotic Analysis . 60 Contents ix 3.3 Cutting Paths . 67 3.3.1 Expansion Operator and Results . 67 3.3.2 Total number of paths . 71 3.4 Cutting Old Leaves . 75 3.4.1 Preliminaries . 75 3.4.2 Expansion Operator and Asymptotic Results . 76 3.5 Cutting Old Paths . 80 3.5.1 Expansion Operator . 80 3.5.2 Analysis of Tree Size and Related Parameters . 83 3.5.3 Total number of old paths . 88 3.6 Future Work . 89 4 Growing and Destroying Catalan–Stanley Trees 91 4.1 Introduction . 91 4.2 Growing Catalan–Stanley Trees . 93 4.3 Age of Catalan–Stanley Trees . 98 4.4 Analysis of Ancestors . 102 5 Ascents in Non-Negative Lattice Paths 107 5.1 Introduction . 107 5.2 Generating Functions: An Analytic Approach . 111 5.3 Generating Functions: A Combinatorial Approach . 115 5.4 Singularity Analysis of Inverse Functions . 120 5.5 Analysis of Ascents . 124 5.5.1 Analysis of Excursions . 124 5.5.2 Analysis of Dispersed Excursions . 127 5.5.3 Analysis of Meanders . 131 Bibliography 137 List of Figures 1.1 Iterated application of a simple deterministic tree reduction operator . .2 1.2 Descendents of the trivial tree in the
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