  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.
