In this thesis our attention was limited to the subject of branching processes where we considered two basic problems of this subject namely the problem of finding the probability of extinction of the offspring of branching processes in general and the problem of finding the offspring distribution (more generally the distribution of the random variables representing the number of offspring in the nth generation). Three distributions were considered for generating the number of the offspring of the nth generation (namely binomial, Poisson, and uniform distributions), by using the stochastic (sometimes called Monte Carlo) simulation. The results of the probability of extinction of the offspring for each generation by the aid of the same above three distributions were studied theoretically and assessed practically by using simulation. Furthermore, the distribution of the offspring in the nth generation is attributed, as a hypothesis, to each of the three considered distributions separately by taking into account that their means equal to a scalar multiple of the distribution mean of the first generation. These hypotheses were tested by C/r/-square goodness of fit test.