Assume that $k$ is even and that $\vec{\imath} \in \mathcal{B}_{k}$ is a cycle passing through $\frac{k}{2} + 1$ distinct vertices (i.e. $|\vec{\imath}| = \frac{k}{2} + 1$). We traverse the edges of $\vec{\imath}$ in order. To each edge of $\vec{\imath}$ we associate an opening parenthesis if this edge appears for the first time and a closing parenthesis if it appears for the second time.
Justify that we thus obtain a well-parenthesized word of length $k$.