Consider a population that is expanding in two-dimensional space. Suppose we collect data from a sample of n individuals taken at random either from the entire population, or from near the outer boundary of the population. A quantity of interest in population genetics is the site frequency spectrum, which is the number of mutations that appear on k of the sampled individuals, for k = 1, . . . , nâ1. As long as the mutation rate is constant, this number will be roughly proportional to the total length of all branches in the genealogical tree that are on the ancestral line of k sampled individuals. While the rigorous literature has primarily focused on models without any spatial structure, in many natural settings, such as tumors or bacteria colonies, growth is dictated by spatial constraints. A large number of such two dimensional growth models are expected to fall in the so-called KPZ universality class, exhibiting similar features as the Kardar-Parisi-Zhang equation. For such models, we adopt the perspective that the genealogical tree can be approximated by the tree formed by the infinite upward geodesics in the directed landscape, a universal scaling limit constructed by Dauvergne, Ortmann, and Virag (2022), starting from n randomly chosen points. This leads to the prediction that the number of mutations inherited by k of the sampled individuals should be proportional to kâ7/5 when the sample comes from the entire population, and proportional to kâ1/2 when the sample comes from the outer edge of the population. The results verify and extend nonrigorous predictions of Fusco, Gralka, Kayser, Anderson, and Hallatschek (2016) and Eghdami, Paulose, and Fusco (2022). This talk is based on joint work with Shirshendu Ganguly and Yubo Shuai.