Approximation of the number of descendants in branching processes

We discuss approximations of the relative limit densities of descendants in Galton–Watson processes that follow from the Karlin–McGregor near-constancy phenomena. These approximations are based on the fast exponentially decaying Fourier coefficients of Karlin–McGregor functions and the binomial coefficients. The approximations are sufficiently simple and show good agreement between approximate and exact values, which is demonstrated by several numerical examples.