We analyze the Na, Mg, Al, and Si abundances in the atmospheres of more than 40 stars, including red giants of different spectral subgroups (normal red giants, mild and classical barium stars) and several supergiants. All these elements exhibit abundance excesses, with the overabundance increasing with the star's luminosity. The dependence of the overabundances for each of these elements on the luminosity (or logg) is the same for all the spectral subgroups, testifying to a common origin: they are all products of hydrogen burning in the NeNa and MgAl cycles that have been dredged up from the stellar interiors to the outer atmospheric layers by convection that gradually develops during the star's evolution from the main sequence to the red-giant stage. The sodium abundances derived for several stars are lower than for other stars with similar atmospheric parameters. The ages and kinematic characteristics of these two groups of stars suggest that they probably belong to different stellar generations.