I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining MO post about awfully sophisticated proofs of simple results got me wondering whether there are any applications of de Branges's theorem (a.k.a. the Bieberbach conjecture) to get estimates in analytic combinatorics. Namely, the hypothetical result would provide a (non-asymptotic) upper bound on a counting sequence by using (at least as a key step) de Branges's Theorem after checking that the associated generating function is related to a univalent mapping from the unit disk.
I'd be interested to see any such nontrivial examples (including entertaining ones that give an awfully sophisticated proof of an estimate that already has a more elementary known proof).
ADDED (09/20/2023): Just to illustrate the idea, here is one application that gives a rough estimate. The function $\tan z$ is univalent in the disk of radius $\pi/2$, so $\frac{2}{\pi} \tan \left( \frac{\pi}{2} z\right)$ is in class $S$ (univalent in the unit disk and normalized to vanish at the origin and have unit derivative at the origin). Applying de Branges's Theorem gives a coefficient bound, and using that $\tan z$ is the exponential generating function for the alternating permutations, this leads to the upper bound $A_n \leq n \left(\frac{2}{\pi}\right)^{n-1} n!$ for the number of alternating permutations $A_n$ of odd size $n$. This estimate is off by an asymptotically linear factor. Since $\tan z$ is odd, we can get a better estimate using the Robertson conjecture (also proved by de Branges). In search of an application with an asymptotically sharp estimate (or at least at the correct order of growth), perhaps an example with a pole of order two should be considered in light of the form taken by the extremal--the Koebe function $z/(1-z)^2$.
