  In this note, we give two combinatorial proofs of the fact that the number of
loopless digraphs on the vertex set $[n]$ with no isolated vertices and with
exactly one Eulerian tour up to a cyclic shift is $\frac{1}{2}(n-1)!C_{n}$,
where $C_{n}$ denotes the $n$-th Catalan number. We construct a bijection with
a set of labeled rooted plane trees and with valid parenthesis arrangements.
