  Trees are useful entities allowing to model data structures and hierarchical
relationships in networked decision systems ubiquitously. An ordered tree is a
rooted tree where the order of the subtrees (children) of a node is
significant. In combinatorial optimization, generating ordered trees is
relevant to evaluate candidate combinatorial objects. In this paper, we present
an algebraic scheme to generate ordered trees with $n$ vertices with utmost
efficiency; whereby our approach uses $\mathcal{O}(n)$ space and
$\mathcal{O}(1)$ time in average per tree. Our computational studies have shown
the feasibility and efficiency to generate ordered trees in constant time in
average, in about one tenth of a millisecond per ordered tree. Due to the 1-1
bijective nature to other combinatorial classes, our approach is favorable to
study 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 use
in devising algorithms for planning and combinatorial optimization involving
Catalan numbers.
