Catalan numbers

Catalan numbers are a sequence of natural numbers that count specific types of combinatorial structures. For example, they tell us how many ways to correctly match pairs of parentheses in an expression, how many different binary tree shapes exist with a given number of nodes, or how many ways to triangulate a polygon. Each Catalan number corresponds to a particular size or complexity of these structures, following a specific formula. They appear in various areas of mathematics and computer science, demonstrating interesting patterns related to recursion and self-similarity.