Abstract.
Let T n denote the set of triangulations of a convex polygon K with n sides. We study functions that measure very natural ``geometric'' features of a triangulation τ∈ T n , for example, Δ n (τ) which counts the maximal number of diagonals in τ incident to a single vertex of K . It is familiar that T n is bijectively equivalent to B n , the set of rooted binary trees with n-2 internal nodes, and also to P n , the set of nonnegative lattice paths that start at 0 , make 2n-4 steps X i of size \(\pm\) 1, and end at X 1 + . . . +X 2n-4 =0 . Δ n and the other functions translate into interesting properties of trees in B n , and paths in P n , that seem not to have been studied before. We treat these functions as random variables under the uniform probability on T n and can describe their behavior quite precisely. A main result is that Δ n is very close to logn (all logs are base 2 ). Finally we describe efficient algorithms to generate triangulations in T n uniformly, and in certain interesting subsets.
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Received August 18, 1997, and in revised form November 5, 1997.
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Devroye, L., Flajolet, P., Hurtado, F. et al. Properties of Random Triangulations and Trees . Discrete Comput Geom 22, 105–117 (1999). https://doi.org/10.1007/PL00009444
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DOI: https://doi.org/10.1007/PL00009444