Schröder parenthesizations and chordates (Q1331129)

\textit{M. Haiman} and \textit{W. Schmitt} [J. Comb. Theory, Ser. A 50, No. 2, 172-185 (1989; Zbl 0747.05007)] and \textit{P. L. Erdős} and \textit{L. A. Székely} [Adv. Appl. Math. 10, 488-496 (1989; Zbl 0723.05046)]\ established a bijection between the set of leaf-labelled trees with \(n\) leaves and \(k\) internal nodes, and the set of partitions of the set \(\{1,2, \dots, n + k - 1\}\) into exactly \(k\) classes. The paper extends the correspondence to leaf-labelled trees in which internal nodes are enriched with a structure.