Let P be a set of n points in the plane. A crossing-free structure on P is a plane graph with vertex set P. Examples of crossing-free structures include triangulations of P, spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of (straight-edge) triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30 n) and at least Ω(2.43 n) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P. For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P. We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of P. The running time of our algorithms is upper bounded by n O(k) , where k i
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