Trees are a fundamental object in graph theory and combinatorics as well as a basic object for data structures and algorithms in computer science. During the last years research related to (random) trees has been constantly increasing and several asymptotic and probabilistic techniques have been developed in order to describe characteristics of interest of large trees in different settings. The aim of this book is to provide a thorough introduction into various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques. It should serve as a reference book as well as a basis for future research. One major conceptual aspect is to bridge combinatorial and probabilistic methods that range from counting techniques (generating functions, bijections) over asymptotic methods (saddle point techniques, singularity analysis) to various sophisticated techniques in asymptotic probability (martingales, convergence of stochastic processes, concentration inequalities). Trees Are A Fundamental Object In Graph Theory And Combinatorics As Well As A Basic Object For Data Structures And Algorithms In Computer Science. During Thelastyearsresearchrelatedto(random)treeshasbeenconstantlyincreasing And Several Asymptotic And Probabilistic Techniques Have Been Developed In Order To Describe Characteristics Of Interest Of Large Trees In Di?erent Settings. Thepurposeofthisbookistoprovideathoroughintroductionintovarious Aspects Of Trees In Randomsettings Anda Systematic Treatment Ofthe Involved Mathematicaltechniques. It Shouldserveasa Referencebookaswellasa Basis For Future Research. One Major Conceptual Aspect Is To Connect Combinatorial And Probabilistic Methods That Range From Counting Techniques (generating Functions, Bijections) Over Asymptotic Methods (singularity Analysis, Saddle Point Techniques) To Various Sophisticated Techniques In Asymptotic Probab- Ity (convergence Of Stochastic Processes, Martingales). However, The Reading Of The Book Requires Just Basic Knowledge In Combinatorics, Complex Analysis, Functional Analysis And Probability Theory Of Master Degree Level. It Is Also Part Of Concept Of The Book To Provide Full Proofs Of The Major Results Even If They Are Technically Involved And Lengthy. Classes Of Random Trees -- Generating Functions -- Advanced Tree Counting -- The Shape Of Galton-watson Trees And Pólya Trees -- The Vertical Profile Of Trees -- Recursive Trees And Binary Search Trees -- Tries And Digital Search Trees -- Recursive Algorithms And The Contraction Method -- Planar Graphs. Michael Drmota. Out of research related to (random) trees, several asymptotic and probabilistic techniques have been developed to describe characteristics of large trees in different settings. The aim here is to provide an introduction to various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques. The aim of this book is to provide a thorough introduction to various aspects of trees in random settings and a systematic treatment of the mathematical analysis techniques involved. It should serve as a reference book as well as a basis for future research.
