🔢 Catalan Numbers

One count, many shapes. Balanced parentheses, Dyck paths, binary trees, polygon triangulations and non-crossing chord diagrams are all counted by the same number Cₙ because there are bijections between them. A "(" is an up-step and a tree's left subtree; a ")" is a down-step and the right subtree. Triangulating an (n+2)-gon by choosing the triangle on a fixed edge splits it exactly like the recurrence Cₙ₊₁ = Σ Cᵢ·Cₙ₋ᵢ. So solving one problem solves all five.