Data from: A Lévy expansion strategy optimizes early dune building by beach grasses. Nature Communications Here, we report on the discovery that heavy-tailed random walks underlie the ability of clonally expanding plants to self-organize and dictate the formation of biogeomorphic landscapes. Using cross-Atlantic surveys, we show that congeneric beach grasses adopt distinct heavy-tailed clonal expansion strategies. Next, we demonstrate with a spatially-explicit model and a field experiment that the Lévy-type strategy of the species building the highest dunes worldwide generates a clonal network with a patch shoot organization that optimizes sand trapping efficiency. This dataset contains data of the survey in which we investigated the st...