The reconstruction of complete vascular trees from medical images has many important applications. Although vessel detection has been extensively investigated, little work has been done on how connect the results to reconstruct the full trees. In this paper, we propose a novel theoretical framework for automatic vessel connection, where the automation is achieved by leveraging constraints from the physiological properties of the vascular trees. In particular, a physiological functional cost for the whole vascular tree is derived and an efficient algorithm is developed to minimize it. The method is generic and can be applied to different vessel detection/segmentation results, e.g. the classic rigid detection method as adopted in this paper. We demonstrate the effectiveness of this method on both 2D and 3D data.