Birkhoff normal form via decorated trees

The paper’s Theorem 4.3 states an explicit decorated-tree formula for the Birkhoff normal form and constructs the generators Fi; its proof proceeds by induction, using a tree-based Taylor/Lie expansion and a collecting identity for nested Poisson brackets. The candidate solution reproduces the same structure: bookkeeping of sizes |T|, the phase-division identity that yields {H0,Fi} cancellation of non-resonant parts, the tree-collecting identity via S(T), a base case, and the inductive step that enforces the ‘forbidden n-node’ constraint in T^{\circ}_{m+2,\ell}. Aside from minor differences in emphasis (Lie-transform notation vs. composition) and a few implicit assumptions (e.g., when the root decoration changes from n to \circ the coefficients match as per the paper’s S-recursion), the arguments agree in substance with the paper’s proof.