Palm vein recognition is an emerging biometric technology that offers enhanced security and privacy. However, acquiring sufficient palm vein data for training deep learning-based recognition models is challenging due to the high costs of data collection and privacy protection constraints. This has led to a growing interest in generating pseudo-palm vein data using generative models. Existing methods, however, often produce unrealistic palm vein patterns or struggle with controlling identity and style attributes. To address these issues, we propose a novel palm vein generation framework named PVTree. First, the palm vein identity is defined by a complex and authentic 3D palm vascular tree, created using an improved Constrained Constructive Optimization (CCO) algorithm. Second, palm vein patterns of the same identity are generated by projecting the same 3D vascular tree into 2D images from different views and converting them into realistic images using a generative model. As a result, PVTree satisfies the need for both identity consistency and intra-class diversity. Extensive experiments conducted on several publicly available datasets demonstrate that our proposed palm vein generation method surpasses existing methods and achieves a higher TAR@FAR=1e-4 under the 1:1 Open-set protocol. To the best of our knowledge, this is the first time that the performance of a recognition model trained on synthetic palm vein data exceeds that of the recognition model trained on real data, which indicates that palm vein image generation research has a promising future.