Topological defect nucleation and boundary branching in crystal growth on a curved surface are two typical elastic instabilities driven by curvature induced stress, and have usually been discussed separately in the past. In this work they are simultaneously considered during crystal growth on a sphere. Phase diagrams with respect to sphere radius, size, edge energy and stiffness of the crystal for the equilibrium crystal morphologies are achieved by theoretical analysis and validated by Brownian dynamics simulations. The simulation results further demonstrate the detail of morphological evolution governed by these two different stress relaxation modes. Topological defect nucleation and boundary branching not only compete with each other but also coexist in a range of combinations of factors. Clarification of the interaction mechanism provides a better understanding of various curved crystal morphologies for their potential applications.