Reinterpreting the Middle-Levels Theorem via Natural Enumeration of Ordered Trees

Let $0<k\in\mathbb{Z}$. A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graph $M_k$ induced by the vertices of the $(2k+1)$-cube representing the $k$- and $(k+1)$-subsets of $\{0,\ldots,2k\}$ is given via an associated dihedral quotient graph of $M_k$ whose vertices represent the ordered (rooted) trees of order $k+1$ and size $k$.