 First-return decomposition of a Dyck path

A nonempty Dyck path starts with $U$. At its first later return to height zero, cut after the matching $D$. The segment between these two steps stays at height at least one; translating it down one level gives a Dyck path $P$. The remaining suffix is another Dyck path $Q$. This gives the unique form $UPDQ$.

If the original semilength of a Dyck path is $n$ and that of $P$ is $k$, then $Q$ has semilength of a Dyck path $n-1-k$. The first return occurs after $2(k+1)$ steps. For the illustrated path UUUDDDUD, $n=4$, $P$ is UUDD, $Q$ is UD, and the first return occurs after six steps.