Reaction-diffusion waves describe diverse natural phenomena from crystal growth in physics to range expansions in biology. Two classes of waves are known: pulled, driven by the leading edge, and pushed, driven by the bulk of the wave. Recently, we examined how demographic fluctuations change as the density-dependence of growth or dispersal dynamics is tuned to transition from pulled to pushed waves. We found three regimes with the variance of the fluctuations decreasing inversely with the population size, as a power law, or logarithmically. These scalings reflect distinct genealogical structures of the expanding population, which change from the Kingman coalescent in pushed waves to the Bolthausen-Sznitman coalescent in pulled waves. The genealogies and the scaling exponents are model-independent and are fully determined by the ratio of the wave velocity to the geometric mean of dispersal and growth rates at the leading edge. Our theory predicts that positive density dependence in growth or dispersal could dramatically alter evolution in expanding populations even when its contribution to the expansion velocity is small. On a technical side, our work highlights potential pitfalls in the commonly-used method to approximate stochastic dynamics and shows how to avoid them.