Matching and analysis of patterns or shapes in the digital plane are of utmost importance in various problems of computer vision and pattern recognition. A digital point set is such a pattern that corresponds to an object in the digital plane. Although there exist several data structures that can be employed for Approximate Point Set Pattern Matching (APSPM) in the real domain, they require substantial modification to support algorithms in the digital domain. To bridge this gap, a novel data structure called "angular treerdquo is proposed, targeting an efficient and error-controllable circular range query in the digital plane. The farthest pair of points may be used as the starting correspondence between the pattern set and the background set. Several classical discrete structures and methodologies of computational geometry, as well as some topological features of circles/discs in digital geometry, have been used in tandem, for successful realization of the proposed APSPM algorithm in the digital plane. The APSPM algorithm based on the angular tree has been implemented and tested on various point sets and the reported results demonstrate the efficiency and versatility of the new data structure for supporting APSPM algorithms.