We propose a novel tree-like curvilinear structure reconstruction algorithm based on supervised learning and graph theory. In this work we analyze image patches to obtain the local major orientations and the rankings that correspond to the curvilinear structure. To extract local curvi-linear features, we compute oriented gradient information using steerable filters. We then employ Structured Support Vector Machine for ordinal regression of the input image patches, where the ordering is determined by shape similarity to latent curvilinear structure. Finally, we progressively reconstruct the curvilinear structure by looking for geodesic paths connecting remote vertices in the graph built on the structured output rankings. Experimental results show that the proposed algorithm faithfully provides topological features of the curvilinear structures using minimal pixels for various datasets.