In "Tree hook length formulae, Feynman rules and B-series", Bradley Jones and Karen Yeats state on pg. 9:
Combinatorial Dyson-Schwinger equations are functional equations with solutions in $H_R[[z]]$ (the Connes-Kreimer Hopf algebra of undecorated rooted trees) using grafting operators, products, inverses, and the empty tree, $I$. As an example consider
$$X(z) = I − zB_{+}(X(z)^{−1})$$
where the inverted series should be expanded as a geometric series.
They then proceed to give the solution in terms of forests of trees in the following line of their paper. The signs and integer coefficients to the order of expansion in the paper agree with those of example III of involutive partition polynomials related to compositional inversion of Laurent series
$$f(z) = a_0 \; z + a_1 + \frac{a_2}{z} + \frac{a_3}{z^2} + \cdots$$
given in my MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution". These partition polynomials are also discussed in OEIS https://oeis.org/A355201.
Question:
Is the tree expansion in Bradley and Yeats' indeed related to the involutive partition polynomials associated with inversion of the Laurent series? If so, explicitly how?
For quick reference from the MO-Q:
Given the compositional inverse pair of Laurent series
$f(z) = a_0 \; z + a_1 + \frac{a_2}{z} + \frac{a_3}{z^2} + \cdots$
and
$f^{(-1)}(z) = b_0 \; z + b_1 + \frac{b_2}{z} + \frac{b_3}{z^2} + \cdots,$
the first few equalities for the involution are
$b_0 = \frac{1}{a_0}$
$b_1 = -\frac{a_1}{a_0}$
$b_2 = -a_2$
$b_3 = -(a_1a_2+a_0a_3)$
$b_4 = - (a_1^2a_2 +a_0a_2^2 + + 2 a_0a_1 a_3+a_0^2 a_4)$
$b_5 = -( a_1^3 a_2+ 3 a_0a_1 a_2^2+ 3 a_0a_1^2 a_3+ 3a_0^2 a_2 a_3+3 a_0^2a_1 a_4 + a_0^3 a_5)$.
