We present a general approach for solving the point-cloud matching problem for the case of mildly nonlinear transformations. Our method quickly finds a coarse approximation of the solution by exploring a reduced set of partial matches using an approach to which we refer to as Active Testing Search (ATS). We apply the method to registration of graph structures by branching point matching. It is based solely on the geometric position of the points, no additional information is used nor the knowledge of an initial alignment. In the second stage, we use dynamic programming to refine the solution. We tested our algorithm on angiography, retinal fundus, and neuronal data gathered using electron and light microscopy. We show that our method solves cases not solved by most approaches, and is faster than the remaining ones.