In Chapter 1, the introduction to the Galton-Watson Branching Process is discussed. The process was first formulated in 1874 by Sir Francis Galton and H.W. Watson to study the evolution of families over generations, marking the first significant application of probability theory to population dynamics. The model involves objects (such as individuals or particles) that produce offspring according to a fixed probability distribution, leading to a tree-like structure where each generation branches off from the previous one. The Galton-Watson Branching Process is defined by a sequence of non-negative integer-valued random variables \(X_n\), where \(X_n\) represents the total number of objects in the \(n\)-th generation. The process is governed by a probability distribution \(\tilde{p}\) on the set of non-negative integers, which remains constant across generations. Each member of a generation produces offspring independently, following the distribution \(\tilde{p}\). Key probabilistic questions about the process include the expected size and variance of the \(n\)-th generation, as well as the probability of eventual extinction (i.e., the probability that \(X_n\) tends to 0 as \(n\) approaches infinity). The initial set of objects is assumed to be fixed, and the process is analyzed to understand its behavior, particularly in terms of finite-time and asymptotic behavior.In Chapter 1, the introduction to the Galton-Watson Branching Process is discussed. The process was first formulated in 1874 by Sir Francis Galton and H.W. Watson to study the evolution of families over generations, marking the first significant application of probability theory to population dynamics. The model involves objects (such as individuals or particles) that produce offspring according to a fixed probability distribution, leading to a tree-like structure where each generation branches off from the previous one. The Galton-Watson Branching Process is defined by a sequence of non-negative integer-valued random variables \(X_n\), where \(X_n\) represents the total number of objects in the \(n\)-th generation. The process is governed by a probability distribution \(\tilde{p}\) on the set of non-negative integers, which remains constant across generations. Each member of a generation produces offspring independently, following the distribution \(\tilde{p}\). Key probabilistic questions about the process include the expected size and variance of the \(n\)-th generation, as well as the probability of eventual extinction (i.e., the probability that \(X_n\) tends to 0 as \(n\) approaches infinity). The initial set of objects is assumed to be fixed, and the process is analyzed to understand its behavior, particularly in terms of finite-time and asymptotic behavior.