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Classic cycle-joining techniques have found widespread application increating universal cycles for a diverse range of combinatorial objects, such asshorthand permutations, weak orders, orientable sequences, and various subsetsof $k$-ary strings, including de Bruijn sequences. In the most favorablescenarios, these algorithms operate with a space complexity of $O(n)$ andrequire $O(n)$ time to generate each symbol in the sequences. In contrast,concatenation-based methods have been developed for a limited selection ofuniversal cycles. In each of these instances, the universal cycles can begenerated far more efficiently, with an amortized time complexity of $O(1)$ persymbol, while still using $O(n)$ space.  This paper introduces $\mathit{concatenation~trees}$, which serve as thefundamental structures needed to bridge the gap between cycle-joiningconstructions based on the pure cycle register and correspondingconcatenation-based approaches. They immediately demystify the relationshipbetween the classic Lyndon word concatenation construction of de Bruijnsequences and a corresponding cycle-joining based construction. To underscoretheir significance, concatenation trees are applied to construct universalcycles for shorthand permutations and weak orders in $O(1)$-amortized time persymbol. Moreover, we provide insights as to how similar results can be obtainedfor other universal cycles including cut-down de Bruijn sequences andorientable sequences.

Concatenation trees: A framework for efficient universal cycle and de Bruijn sequence constructions