Schröder parenthesizations and chordates

We establish that Schroder trees are a subclass of Schröder parenthesizations by a natural bijection. The Haiman-Schmitt bijection between Schröder parenthesizations, enriched by uniform species and partitions, generalizes to a bijection between Schröder parenthesizations and assemblies. Using these bijections, we prove some tree counting formulas. We also introduce the definitions of trees over a partition and similarly chordates over a partition. These structures give rise to some beautiful enumeration formulas.