TITLE: Combinatorial interpretation of series reversion coefficients
QUESTION [8 upvotes]: In a recent paper studying some generalizations of Stirling numbers, my coauthors and I needed the following result:
If $f(x)=\sum_{n \geq 1} a_n x^n/n!$ is a power series with $a_1 \neq 0$, and $g(x) = \sum_{n \geq 1} b_n x^n/n!$ is its series reversion (so $f(g(x))=g(f(x))=x$), then
$$
b_n = \sum_{T} w(T)
$$
where the sum is over phylogenetic trees on $\{1, \ldots, n\}$. Here a phylogenetic tree on $\{1,\ldots, n\}$ is a rooted tree with $n$ leaves, no vertex with exactly one child, and leaves labelled from the set $\{1, \ldots, n\}$ (but no other vertex receiving a label); see page 3 of the paper linked above for a picture. The weight $w(T)$ is computed as follows: if $T$ has $n$ leaves and $m$ non-leaves then
$$
w(T) = (−1)^m a_1^{-(m+n)} \prod \{a_{d(v)}: v~\mbox{a non-leaf}\}
$$
where $d(v)$ denotes the number of children of $v$ (and when $n=1$ the isolated vertex is considered a leaf).
The proof is short, and this feels like something that should have been written down prior to now, but we could find no reference.
Question: Has the above result appeared in the literature?
(There is a paper of Chen from 1990 with a similar result but with a distinctly different and not obviously equivalent formulation. Chen shows that $b_n$ is a weighted sum over Schroeder trees: trees on $n$ labelled vertices (as opposed to leaves), with no restriction on numbers of children, in which each non-leaf is endowed with an ordered partition of its children. The count of Schroeder trees by number of vertices is A053492. The count of phylogenetic trees by number of leaves is A000311.)

REPLY [6 votes]: This result can also be found as Theorem 3.3.9 in Ginzburg-Kapranov: "Koszul duality for operads" (Duke Math J 1994). Set $r=1$ in their result and rescale so that $a_1=1$. They had discovered it independently, but thank D. Wright for informing them that the result was known before due to J. Towber. They also give a reference to Wright: "The tree formulas for reversion of power series" (JPAA 1989), where this result appears as Theorem 3.10.
The context of this result in the setting of operads is that any dg operad $P = \{P(n)\}$, say with $P(0)=0$ and $P(1)$ a copy of the base field, has a generating series which encodes the Euler characteristics of the chain complexes $P(n)$. The generating series of $P$ is the compositional inverse of the generating series for the "bar construction" on the operad $P$. The bar construction of operads is defined by a sum over trees whose internal vertices are decorated by the chain complexes of the operad $P$; in fact, the trees involved are exactly those that you call phylogenetic trees.