In probability theory, a **branching process** is a type of discrete‐time stochastic process that models the evolution of a population in which each individual in generation $n$ independently produces some random number of “offspring” in generation $n+1$.  A common example is the **Galton–Watson process**. # Definitions > [!NOTE] Definition (ST227) > Let $X_0 \in \mathbb{N}$. Let $Y$ be a [[Random Variables|random variable]] such that $\mathbb{P}(Y \in \mathbb{N} \cup\{0\})=1$ (state space is $\mathbb{N} \cup \{ \}$). Let $\left(X_n\right)_{n \geq 0}$ be a [[Stochastic Process|stochastic process]] such that for every $n \in \mathbb{N} \cup\{0\}$ > > $ > \begin{array}{ll} > X_{n+1}=\sum_{k=1}^{X_n} Y_{n k} & \text { if } X_n>0 \\ > X_{n+1}=0 & \text { if } X_n=0 > \end{array} > $ > > where $\left\{Y_{i j}, i \in \mathbb{N} \cup\{0\}, j \in \mathbb{N}\right\}$ are independent copies of $Y\left(Y_{i j} \sim Y, i \in \mathbb{N} \cup\{0\}\right.$, $j \in \mathbb{N}$ ). > > Then $\left(X_n\right)_{n \geq 0}$ is called a *branching process* with the *offspring distribution* $Y$. # Properties