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We introduce a new multiscale point cloud representation scheme based on a tree-structured multiscale piecewise plane approximation of the underlying 3D surface geometry. Dyadic division and plane approximation of points within each dyadic cube result in a tree-structured multiscale representation of the 3D surface described by point cloud data. We then perform a complexity-regularized tree pruning combined with adaptive control of plane thickness to obtain a very compact representation of the surface geometry. Based on its adaptivity and multiscale structure, the proposed representation scheme provides a desirable framework for modelling, coding and processing of point cloud data. We demonstrate the performance of the proposed algorithm through experiments of rendering point clouds.