Objective: Intraoperative liver deformation poses a considerable challenge during liver surgery, causing significant errors in image-guided surgical navigation systems. This study addresses a critical non-rigid registration problem in liver surgery: the alignment of intrahepatic vascular trees. The goal is to deform the complete vascular shape extracted from preoperative Computed Tomography (CT) volume, aligning it with sparse vascular contour points obtained from intraoperative ultrasound (iUS) images. Challenges arise due to the intricate nature of slender vascular branches, causing existing methods to struggle with accuracy and vascular self-intersection. Methods: We present a novel non-rigid sparse-dense registration pipeline structured in a coarse-to-fine fashion. In the initial coarse registration stage, we introduce a parametrization deformation graph and a Welsch function-based error metric to enhance convergence and robustness of non-rigid registration. For the fine registration stage, we propose an automatic curvature-based algorithm to detect and eliminate overlapping regions. Subsequently, we generate the complete vascular shape using posterior computation of a Gaussian Process Shape Model. Results: Experimental results using simulated data demonstrate the accuracy and robustness of our proposed method. Evaluation results on the target registration error of tumors highlight the clinical significance of our method in tumor location computation. Comparative analysis against related methods reveals superior accuracy and competitive efficiency of our approach. Moreover, Ex vivo swine liver experiments and clinical experiments were conducted to evaluate the method's performance. Conclusion: The experimental results emphasize the accurate and robust performance of our proposed method. Significance: Our proposed non-rigid registration method holds significant application potential in clinical practice.