In a simple branching process we assume that each individual produces a number of offspring, who in turn produce their own offspring. The process always starts with one individual (the #~{ancestor}) which is denoted generation 0. The offspring of the ancestor are called the 1^{st} generation, and in general the offspring of the ~n^{th} generation are called the ~n+1^{th} generation. (Note that in the simple model we take no account of the time between generations.)
We make the following assumptions in the model we are dealing with:
This is known as the #~{Gaton-Watson} model for a branching process.
Let ~X denote the number of offspring of an individual, and put _ ~p( ~x ) _ = _ ~P ( ~X = ~x ) , _ called the #~{offspring probability function} of the individual. We have _ &sum._{~x}^{&infty.}_{= 0} _ ~p( ~x ) _ = _ 1.
The number of individuals in the ~n^{th} generation is denoted ~Z_~n. We have _ ~Z_0 = 1 , _ ~Z_1 = ~X , _ and
~Z_~n _ = _ sum{~X_~i^{( ~n - 1 )},~i = 1,~Z_{~n - 1}} _ _ _ _ ~n > 1
Where _ ~X_~i^{( ~n - 1 )} is the number of offspring of the ~i^{th} individual in the ~n - 1^{th} generation.