Catalan's identity

Catalan's identity is a mathematical formula that links different ways of counting certain structures, like binary trees or balanced parentheses. It expresses that the number of these structures with a specific number of components can be calculated using binomial coefficients and previous counts. Essentially, it states that each count can be broken down into a sum of products of smaller counts, reflecting the recursive nature of these structures. This identity helps mathematicians understand how complex arrangements relate to simpler ones, providing a foundation for counting and analyzing various combinatorial objects efficiently.