In a recent [paper](https://arxiv.org/abs/1610.05803) 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](https://pdfs.semanticscholar.org/b079/4b05709432e882698c47fbeac625d291743d.pdf) 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](http://oeis.org/A053492). The count of phylogenetic trees by number of leaves is [A000311](http://oeis.org/A000311).) 



Combinatorial interpretation of series reversion coefficients
Yes, this is known. As Tom Copeland mentioned, this is in [Drake's thesis](http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf), Example 1.4.7.


Drake gives a broad generalization of your result. His main theorem (Theorem 1.3.3) is roughly as follows. Suppose you have an alphabet $A$ of certain trees, and a set $L$ of 'allowed links' between them, ways of connecting the root of one to a leaf of another. Let $S(L,n)$ be the set of all trees $T$ with $n$ leaves labeled bijectively $1, \ldots, n$ with no other nodes labeled with the property that $T$ made up of the letters $A$ with *only* links between them which are in $L$. Say each tree $T \in A$ has a weight $w(T)$, and a tree made up of letters $T\_1, \ldots, T\_n \in A$ with certain links between them has weight $w(T\_1) \cdots w(T\_n)$. Let $$F(x) = \sum\_{n=1}^\infty a\_n \frac{x^n}{n!}$$ where $a\_n = \sum\_{T \in S(L,n)} w(T)$. Then the compositional inverse is $$F^{-1}(x) = \sum\_{n=1}^\infty b\_n \frac{x^n}{n!}$$ where $$b\_n = \sum\_{T \in S(\bar{L},n)} (-1)^{m(T)} w(T).$$
Here $m(T)$ is the number of letters in $A$ that $T$ is composed of, and $\bar{L}$ is the complement of $L$ in the set of all possible links between elements of $A$.


This was an extension of the work of Susan Parker, who had a similar theorem using ordinary generating functions. Parker's result can, in turn, be seen as a generalization of a result of Carlitz, Scoville and Vaughan. Ira Gessel's [slides](http://people.brandeis.edu/~gessel/homepage/slides/csv.pdf) may be helpful.


To get your theorem, let $A$ be the set of "simple" trees, the trees $T\_n$, $n > 1$, so that $T\_n$ is a tree with the root having the $n$ leaves $1,2, \ldots, n$ as children. Let $w(T\_n) = t\_n$ (an indeterminate). Let $L$ be the empty set of links. Then $$F(x) = x + t\_2\frac{x^2}{2!} + t\_3 \frac{x^3}{3!} + \cdots$$ since $S(L,n) = \{T\_n\}$ for $n>1$, and $S(L,1)$ consists of the tree with one node which is always allowed (and has weight $1$).


Then by Drake's inversion theorem, $$F^{-1}(x) = \sum\_{n=1}^\infty b\_n \frac{x^n}{n!}$$ where $$b\_n = \sum\_{T \in S(\bar{L},n)} (-1)^{m(T)} w(T)$$ is the sum of the weights of *all* phylogenetic trees (since $\bar{L}$ is the set of all links.)


See also [my thesis](https://digital.lib.washington.edu/researchworks/handle/1773/36757), which contains a different generalization of your result. The phylogenetic trees (also called total partitions) are mentioned in Section 3.2


