Pattern Recognition Letters 12 (1991) 139-144 March 1991 North-Holland Counting thin and convex polygons bushy triangulations of S. Chattopadhyay Department o f Computer Science and Engineering, Jadavpur University, Calcutta-700032, India P.P. Das Department of Computer Science and Engineering, Indian Institute of Technology, Kharagpur-721302, India Received 7 February 1990 Revised 18 December 1990 A bstract Chattopadhyay, S. and P.P. Das, Counting thin and bushy triangulations of convex polygons, Pattern Recognition Letters 12 (1991) 139-144. Triangulation of a simple polygon is a classical problem of immense interest in computational geometry and related fields. Recently, some specialized kinds of triangulations, namely thin and bushy triangulations have received attention owing to their use in pattern recognition and in finding geodesic properties. As triangulation of a given polygon is essentially non-unique, counting the number of ways a polygon can be triangulated is an interesting problem. In this paper, we solve two such counting problems, to find ~t(n) (t~b(n)), the number of thin (bushy) triangulations of an n-sided convex polygon and propose many challenging ones. These numbers provide exact upper bounds for the corresponding triangulation and may be used in analyzing average case performance of different triangulation algorithms. Keywords. Computational geometry, triangulation, partitioning. 1. Introduction Triangulation of a simple polygon (where edges intersect at vertices only) is a well-known problem, studied intensively in computational geometry and related fields [2, 3]. It defines a partitioning of the polygon into disjoint triangles (though they share common edges) produced through the insertion of non-intersecting internal diagonals. Clearly, if the polygon has n vertices (sides), (n - 3) diagonals are drawn to triangulate it into (n-2) triangles (let us call them the component triangles). It is easy to see that if n > 3, then no component triangle can share all its three sides with the original polygon. So, component triangles can be classified into degree 2, 1 or 0 depending on whether they share 2, 1 or 0 sides with the original polygon. Component triangles of degree 2 are called the ears of the triangulation. Since the internal diagonals can be drawn in a polygon in a number of possible ways, the triangu- lation of a polygon is necessarily non-unique. It is an interesting idea to count the number of ways J(n) in which a polygon (n-gon) can be triangulated. 0167-8655/91/$03.50 © 1991 -- Elsevier Science Publishers B.V. (North-Holland) 139