The mechanisms of strain relief in submonolayer heteroepitaxy of Cu/Ni(100) are studied using variable temperature scanning tunneling microscopy and high resolution low energy electron diffraction. It is demonstrated that pseudomorphic copper islands, as they grow in size, undergo a spontaneous shape transition. Below a critical island size of about 500 atoms the islands have a compact shape, while above this size they become ramified. The shape transition of the coherently strained islands, predicted theoretically by Tersoff and Tromp, is driven by the size-dependent outward relaxation of the step edge atoms due to the positive lattice mismatch. The ramified island shape, which reflects the energy minimum of binding and strain energy, is characterized by only one parameter: the arm width of the monolayer-high copper islands w = (22 +/- 1) atoms.