Abstract
Polygon triangulations and their generalizations to (m + 2)—angulations are fundamental in combinatorics and computational geometry. This paper presents a unified linear-time framework that establishes explicit bijections between m—Dyck words, planted (m + 1)—ary trees, and (m + 2)—angulations of convex polygons. We introduce stack-based and tree-based algorithms that enable reversible conversion between symbolic and geometric representations, prove their correctness and optimal complexity, and demonstrate their scalability through extensive experiments. The approach reveals a hierarchical decomposition encoded by Fuss–Catalan numbers, providing a compact and uniform representation for triangulations, quadrangulations, pentangulations, and higher-arity angulations. Experimental comparisons show clear advantages over rotation-based methods in both runtime and memory usage. The framework offers a general combinatorial foundation that supports efficient enumeration, compressed representation, and extensions to higher-dimensional or non-convex settings.