Dyck words are counted by Catalan numbers

Statement

Let $n$ be a natural number. Then the number of Dyck words of semilength $n$ is the Catalan number $C_n$.

Remarks

For $n = 3$ the $5$ Dyck words are $UUUDDD$, $UUDUDD$, $UUDDUD$, $UDUUDD$ and $UDUDUD$. Proof idea: a nonempty Dyck word has a unique form $U w_1 D w_2$, where $w_1$ and $w_2$ are Dyck words. This gives the Catalan recurrence.

Explanations

L1 human

Each nonempty balanced bracket string starts with an opening bracket, and that bracket has a matching closing bracket. The inside and the part after the match are smaller balanced strings, so the count satisfies the Catalan recurrence.