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A subfamily of Dyck words called tight Dyck words is seen to correspond, viaa "castling" procedure, to the vertex set of an ordered tree $T$. From $T$, a"blowing" operation recreates the whole family ol Dyck words. The vertices of$T$ can be elementarily updated all along $T$. This simplifies anedge-supplementary arc-factorization view of Hamilton cycles of odd andmiddle-levels graphs found by T. M\"utze et al. This take into account that theDyck words represent: {\bf(a)} the cyclic and dihedral vertex classes of oddand middle-levels graphs, respectively, and {\bf(b)} the cycles of their2-factors, as found by T. M\"utze et al.

Castling tree of tight Dyck nests with applications to odd and middle-levels graphs