Properties of Random Triangulations and Trees

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.