Limit theorems for the total number of descendants for the Galton–Watson branching process

The main results of the present paper deal with the asymptotic behavior of the conditional distribution for the whole number of descendants $S_n $ of a single particle in the Galton–Watson process with respect to the condition that the process degenerates at time n and the expectation for the number of particles generated by one particle tends to 1 as $n \to \infty $.