Data from: A Lévy expansion strategy optimizes early dune building by beach grasses. Nature CommunicationsHere, 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 step size distribution of both beach grasses used in our study (Ammophila arenaria & Ammophila breviligulata). The dataset consists of images, shoot coordinates and distances between shoots (using nearest neighbour methods). In addition we included the data on nutrient levels of both soil and plant tissue taken during the survey. The survey was carried out from April to August 2017 on both the eastern US coast (North Carolina and Virginia) and the Dutch coast (Schiermonnikoog).Next to the survey data we included data of an experiment in which we measured the volume of sand and the sand trapping efficiency of various patterns of dune grass mimics. The experiment was conducted at Schiermonnikoog in the summer of 2016.