The set of notes Mathemagical Forests is an expansion of the May notes and discusses some connections between rooted trees, derivative operators, Lagrange inversion, the Legendre transformation, the Faa di Bruno formula, Sheffer sequences and umbral calculus, and the infinite dimensional Witt Lie algebra.

Added 2/2014: For some early history on operators of the form see page 13 in The Theory of Linear Operators … (Principia Press, 1936) by Harold T. Davis.

Added 12/2016: Connections to pre- and post-Lie algebras and numerical analysis presented in “What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations” by McLachlan, Modin, Munthe-Kaas, and Verdier

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