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Trees are a fundamental object in graph theory and combinatorics as well as a basic object for data structures and algorithms in computer science. This book provides an introduction into various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques.

Trees (Graph theory). --- Trees (Graph theory) --- Stochastic processes --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Stochastic processes. --- Random processes --- Mathematics. --- Data structures (Computer science). --- Algebra. --- Algorithms. --- Probabilities. --- Discrete mathematics. --- Combinatorics. --- Discrete Mathematics. --- Probability Theory and Stochastic Processes. --- Data Structures. --- Graph theory --- Probabilities --- Distribution (Probability theory. --- Data structures (Computer scienc. --- Algorism --- Arithmetic --- Combinatorics --- Mathematical analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Foundations --- Information structures (Computer science) --- Structures, Data (Computer science) --- Structures, Information (Computer science) --- Electronic data processing --- File organization (Computer science) --- Abstract data types (Computer science) --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis

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Number theory --- Algebra --- Operational research. Game theory --- Discrete mathematics --- Computer science --- algebra --- discrete wiskunde --- stochastische analyse --- database management --- programmatielogica --- algoritmen --- kansrekening --- getallenleer

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Lossless data compression is a facet of source coding and a well studied problem of information theory. Its goal is to find a shortest possible code that can be unambiguously recovered. Here, we focus on rigorous analysis of code redundancy for known sources. The redundancy rate problem determines by how much the actual code length exceeds the optimal code length. We present precise analyses of three types of lossless data compression schemes, namely fixed-to-variable (FV) length codes, variable-to-fixed (VF) length codes, and variable to- variable (VV) length codes. In particular, we investigate the average redundancy of Shannon, Huffman, Tunstall, Khodak and Boncelet codes. These codes have succinct representations as trees, either as coding or parsing trees, and we analyze here some of their parameters (e.g., the average path from the root to a leaf). Such trees are precisely analyzed by analytic methods, known also as analytic combinatorics, in which complex analysis plays decisive role. These tools include generating functions, Mellin transform, Fourier series, saddle point method, analytic poissonization and depoissonization, Tauberian theorems, and singularity analysis. The term analytic information theory has been coined to describe problems of information theory studied by analytic tools. This approach lies on the crossroad of information theory, analysis of algorithms, and combinatorics.

Data compression (Computer science)

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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.

Number theory --- Algebra --- Operational research. Game theory --- Discrete mathematics --- Computer science --- algebra --- discrete wiskunde --- stochastische analyse --- database management --- programmatielogica --- algoritmen --- kansrekening --- getallenleer

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Combinatorial analysis --- Graph theory --- Branching processes --- Computer science