Problem A5

A Dyck n-path is a lattice path of n upsteps (1,1) and n downsteps (1,–1) that starts at the origin O and never dips below the x-axis. A return is a maximal sequence of contiguous downsteps that terminates on the x-axis. For example, the Dyck 5-path illustrated has two returns, of length 3 and 1 respectively. 

Show that there is a one-to-one correspondence between the Dyck n-paths with no return of even length and the Dyck (n – 1)-paths.