Since there is exactly one parent at the beginning of the process (i.e. ) who produces all the future children, we can say that the probability that is equal to 1 is 100%, or . Because there is definitely only one parent in the beginning, cannot be equal to any other value except one. This can be written mathematically as for . The probability generation function of the total number of people in generation (i.e. ) is given by:
By plugging in this initial probability observation we see that
Thus we get the boundary condition . If we further define as the probability generation function for the children, , in a particular family of the lineage in an analogous fashion, we have:
From here it then becomes clear there is a relation between and at the beginning generations. In fact, the constraint is that the total number of people in the first generation is determined by the number of children in the family the original parent, , had produced. So the probability generation function of the first generation, is determined by the probability generation function for the children. Thus, .
Since there could be multiple families for any given generation (except the 0th generation) then the number of children, , in a family might be different. To keep track of the possibly different number of children in each family, we redefine as where is the family and is the size of that family. This can be generalized as a function of the generation by also realizing that the total number of children from all families is equal to the size of the entire generation, . In other words:
Since all are independent and identically distributed random variables, then is the sum of a random number of independent and identically distributed random variables. This is important because the following relation holds for this case: