TITLE: The non-convergence of f(f(x))=exp(x)-1 and labeled rooted trees
QUESTION [7 upvotes]: This question is closely related to MO f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential. Consider $e^{e^x-1}$, this is the generating function of the Bell numbers. A more general way to look at Bell numbers is as rooted trees, hierarchies of height 2. Given $g(x)=e^x-1$, $g^n(x), n \in \mathbb{N}$ is the generating function of hierarchies of height n. See page 107 - 110 of Analytic Combinatorics. The ECS should have the integer sequences associated with hierarchies of different heights. Also see OEIS 

    Integer sequence                      height OEIS
    {1,1/2,1/8,0,1/32,-7/128,1/128,159/256}  1/2 A052122
    {0,1,1,1,1,1,1,1,1}                        1
    {1,2,5,15,52,203,877,4140}                 2 A000110
    {1,3,12,60,358,2471,19302,167894}          3 A000258
    {1,4,22,154,1304,12915,146115,1855570}     4 A000307
    {1,-1,2,-6,24,-120,720,-5040}             -1 A000142
    {1,-2,7,-35,228,-1834,17582,-195866}      -2 A003713 

Several solutions for $f(f(x))=e^x-1$ have been proposed on MO, but the work of I.N. Baker is cited as proving that $f(x)$ has no convergent solution, "even in an ϵ-ball around 0." I am currently trying to read the original German, to understand Baker's proof.
Question 1 Could someone summarize Baker's proof? It is frequently referred to and an explanation in English would be wonderful. 
Question 2 Formal power series can contain useful information, even if the are divergent. It seems that divergent series are not treated with quite the contempt they used to be. I believe on the Tetration Forum that someone raised the possibility of $f(x)$ being Borel summable. What are the potential options for "rehabilitating" a series that is not nicely convergent.
Question 3 If $g(x)=e^x-1$, $g^n(x), n \in \mathbb{N}$ is the generating function of hierarchies of height n, doesn't $g(x)=e^x-1$, $g^n(x), n \in \mathbb{R}$ consists of labeled rooted trees of fractional height? So shouldn't $f(x)=g^\frac{1}{2}(x)$ be the generating function for labeled rooted trees of height $\frac{1}{2}$?
Doesn't the divergence of $f(x)=g^\frac{1}{2}(x)$ imply that a label rooted tree of height $\frac{1}{2}$ have infinitely many leaves, that the width of the tree is infinite. Can't be use the fact that we are working with a labeled rooted tree to constrain the width of the tree from becoming infinite?

REPLY [2 votes]: This is an attempt to summarize some work related to question 2 which I do not fully understand myself. I am summarizing Sections 1.1 and 1.2 of Dudko's thesis, whose exposition is excellent (including work of earlier authors which he describes).
Set $F(z) = e^z-1$, so $F(z) = z+z^2/2+z^3/6+O(z^4)$. This discussion will apply to functional square roots of any $F(z)$ of the form $z+c z^2+O(z^3)$ for $c \neq 0$. Set $f(w) = 1/F(1/w)$, so $f(w) = w - 1/2 + w/12 + O(w^2)$. We will attempt to find a composition square root $f^{\langle 1/2 \rangle}(w)$ for $w$, the change of coordinates $w \mapsto 1/z$ will then change it into a compositional square root for $F$. 
Suppose that we had an invertible holomorphic function $\alpha$ obeying
$$\alpha(f(w)) = \alpha(w)-1/2. \quad (\ast)$$
For now, I'll be sloppy about on what region $\alpha$ is defined; this will eventually be a crucial issue. Such an $\alpha$ is called a Fatou coordinate.
Then we could define fractional compositions $f^{\langle s \rangle}$ by $f^{\langle s \rangle}(w) = \alpha^{-1}(\alpha(w)-s/2)$ and we would clearly have $f^{\langle s \rangle} \circ f^{\langle t \rangle} = f^{\langle s+t \rangle}$ and $f^{\langle 1 \rangle}=f$.
There is a unique formal power series solution
$$\alpha(w) = w+\frac{1}{6} \log w + \sum_{n \geq 1} c_n w^{-n}$$
to $(\ast)$.
Dudko shows (Theorem 37) that, for any $\delta>0$ there is an $R>0$ such that the sum $\sum c_n z^{-n}$ is Borel summable on a region of the form $U_+ = \{ r e^{i \theta}: r > R, \theta \in (-\pi+\delta, \pi - \delta) \}$ and is separately Borel summable on a region of the form  $U_- = \{ r e^{i \theta}: r > R, \theta \in (\delta, 2 \pi - \delta) \}$. Here the integral defining the first Borel sum is along the positive real axis, and the integral for the second is on the negative real axis. However, the two Borel summations have different values! I'm not sure how to translate that Borel summability into the Borel summability you are looking for, but it seems in the same neighborhood.