Intrigued by a growing body of research on convective organisation, this study investigates the morphology of precipitating marine cumulus convection with and without cold pools under vertical wind shear. We ran idealised large-eddy-simulation experiments with zonal forward and backward shear and without shear. Without (or only weak) subcloud-layer shear, conditions are unfavourable for convective deepening, as clouds remain stationary relative to their subcloud-layer roots, and precipitative downdrafts interfere with emerging updrafts. Conversely, under forward shear, where the wind strengthens with height (a condition that is commonly found in the trades), clouds move at a faster speed than their roots, and precipitation falls downwind away from emerging updrafts. This significantly facilitates convective deepening, precipitation and consequently the formation of cold pools. Forward shear has another advantage as opposed to weak or backward shear: The existing background vorticity interacts with the (opposing) vorticity of cold-pool gust fronts which facilitates forced uplift. Inhibiting cold-pool formation delays convective deepening only shortly.