This is called a Galton-Watson tree. From one ancestor (the root of the tree), we have a random number $\displaystyle Z_1$ of children, each of which itself has random numbers of children, etc., where the numbers of children are all independent of each other. This is a genealogic tree where the numbers of children is random: k children (possibly 0) with probability $\displaystyle p_k$. A basic question is: does the family eventually get extinct? i.e. is the tree finite?

In this case, $\displaystyle \Omega$ would be the set of trees (or a larger set), or an equivalent representation of trees. A convenient choice is to let $\displaystyle U=\cup_n \mathbb{N}^n$, the set of all finite sequences of integers, and $\displaystyle \Omega=\mathbb{N}^U$, the set of positive-integer sequences indexed by elements of $\displaystyle U$. Intuitively, an individual corresponds to a sequence $\displaystyle u=a_1a_2\cdots a_n\in U$ if it is obtained from the root as the $\displaystyle a_n$-th child of the $\displaystyle a_{n-1}$-th child of .... of the $\displaystyle a_1$-th child of the root. And the number of children of this individual is encoded in the index $\displaystyle n_u$ of the tree $\displaystyle T=(n_u)_{u\in U}\in \mathbb{N}^U$ (with $\displaystyle n_u$ arbitrary if $\displaystyle u$ is not connected to the root, thus there are more codings than trees).