We present a method to find globally optimal topology and trajectory jointly for planar linkages. Planar linkage structures can generate complex end-effector trajectories using only a single rotational actuator, which is very useful in building low-cost robots. We address the problem of searching for the optimal topology and geometry of these structures. However, since topology changes are non-smooth and non-differentiable, conventional gradient-based searches cannot be used. We formulate this problem as a mixed-integer convex programming (MICP) problem, for which a global optimum can be found using the branch-and-bound (BB) algorithm. Compared to existing methods, our experiments show that the proposed approach finds complex linkage structures more efficiently and generates end-effector trajectories more accurately.