The cheapest and thus widespread way to add new generators to a high-voltage power grid is by a simple tree-like connection scheme. However, it is not entirely clear how such locally cost-minimizing connection schemes affect overall system performance, in particular the stability against blackouts. Here we investigate how local patterns in the network topology influence a power grid’s ability to withstand blackout-prone large perturbations. Employing basin stability, a nonlinear concept, we find in numerical simulations of artificially generated power grids that tree-like connection schemes—so-called dead ends and dead trees—strongly diminish stability. A case study of the Northern European power system confirms this result and demonstrates that the inverse is also true: repairing dead ends by addition of a few transmission lines substantially enhances stability. This may indicate a topological design principle for future power grids: avoid dead ends.