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Trees are useful entities allowing to model data structures and hierarchicalrelationships in networked decision systems ubiquitously. An ordered tree is arooted tree where the order of the subtrees (children) of a node issignificant. In combinatorial optimization, generating ordered trees isrelevant to evaluate candidate combinatorial objects. In this paper, we presentan algebraic scheme to generate ordered trees with $n$ vertices with utmostefficiency; whereby our approach uses $\mathcal{O}(n)$ space and$\mathcal{O}(1)$ time in average per tree. Our computational studies have shownthe feasibility and efficiency to generate ordered trees in constant time inaverage, in about one tenth of a millisecond per ordered tree. Due to the 1-1bijective nature to other combinatorial classes, our approach is favorable tostudy the generation of binary trees with $n$ external nodes, trees with $n$nodes, legal sequences of $n$ pairs of parentheses, triangulated $n$-gons,gambler's sequences and lattice paths. We believe our scheme may find its usein devising algorithms for planning and combinatorial optimization involvingCatalan numbers.

An Efficient Scheme for the Generation of Ordered Trees in Constant Amortized Time