Maybe not so surprising.

***Example 1) Flows, the geometry of associahedra, and moduli spaces for marked Riemann sufaces of genus 0***

**A) [Flows][1], [streamlines][2], [integral curves][3], and compositional inversion**:

Let the inverse of the formal power series $h(z)=a_1\:z+ a_2 \: z^2+ \cdots$ be $h^{-1}(z)=b_1 \: z+ b_2 z^2 + \cdots$ ; then, with $\omega=h(z)$  and  $g(z)=1/[dh(z)/dz]$, a flow field is generated by

$$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]z = \exp \left[ {t\frac{d}{{d\omega }}} \right]{h^{ - 1}}(\omega ) = {h^{ - 1}}[t + \omega] = {h^{ - 1}}[t + h(z)]=W(t,z),$$ 

and it is easy to show that the flow map has the following features;

$$<Identity>\:\:\: W(0,z)= z $$

$$<Orbit>\:\:\: W(t,0)= h^{(-1)}(t)$$
$$$$

$$<Velocity/generator>\:\:\: \frac{dW(0,z)}{dt} = g(z) = [h^{(-1)}]^{'}(z)$$ 

$$<Autonomous\:\: ODE>\:\:\: g(h(z)) = [h^{(-1)}]^{'}(h(z))$$

$$<Group\:\:property>\:\:\: W[s,W(t,z)] = W(s+t,z)$$ 

$$<Tangency>\:\:\left [\frac{d}{dt}-g(z)\frac{d}{dz}  \right ]\:W(t,z) = 0,$$

so $(1,-g(z))$ are the components of a vector orthogonal to the gradient of $W$ and, therefore, tangent to the contour of $W$ at $(t,z)$. 

**B) Compositional (Lagrange) inversion and associahedra (cf. [Loday][4])**:

The iterated derivatives acting on $z$ and evaluated a $z=0$ generate the coefficients of the inverse power series. E.g.,

$$b_5=\frac{1}{5!}[g(z)\frac{d}{{dz}}]^{5}z|_{z=0} = \frac{1}{a_1^{9}}  [14\: a_2^{4} - 21\: a_1 a_2^2 a_3 + a_1^2[6  \:a_2 a_4+ 3\:  a_3^2] - 1\: a_1^3 a_5],$$

This is related to a refined f-vector (face-vector) for the 3-D Stasheff polytope, or 3-D associahedron, with **14** vertices (0-D faces), **21** edges (1-D faces), **6** pentagons (2-D faces), **3** rectangles (2-D faces), **1** 3-D polytope (3-D faces). Subtracting two from the index of $a_n$, and ignoring the resulting indeterminates with indices with values less than one, allows one to read off the geometry of the associahedron from cartesian products of the lower dimensional associahedra (Loday), e.g., $3\: a^2_3$ becomes $3\: a^2_1$, the cartesian product of the 1-D associahedron with itself, which is a tetragon, or square in some reps.  

This correspondence between the refined f-vectors of the $n$-D associahedron and $b_{n+2}$ holds in general, (see [OEIS-A133437][5]).

**C) Associahedra and marked Riemann surfaces of genus 0**:

Brown and Bergstrom in "[Inversion of series and the cohomology of the moduli spaces of $M_{0,n}^\delta][6]$" state:

For $n \geq 3$, let $M_{0,n}$ denote the moduli space of genus $0$ curves with $n$ marked points, and $\overline{M}_{0,n}$ its smooth compactification. ... In this paper, we prove that the inverse of the ordinary generating series for the Poincare polynomial of $H^\bullet(M_{0,n})$ is given by the corresponding series for $H^\bullet(M^{\delta}_{0,n})$, where $\overline{M}_{0,n}\subset M^{\delta}_{0,n} \subset \overline{M}_{0,n}$ is a certain smooth affine scheme.

And on page 3, they give the abbreviated formula 

$$M^{\delta}_{0,n}=14\; M_{0,3} \cup 21\: M_{0,4} \cup [6\: M_{0,5} \cup 3\: M^2_{0,4}] \cup M_{0,6}.$$

So, we have a connection between flows determined by the combinatorics of the associahedra and moduli spaces. 

***Example II) The inviscid [Burgers-Hopf equation][7] and associahedra***

Define $$U(x,t)=\frac{x-A(x,t)}{t}$$ and $$A^{-1}(x,t)=x+t\;F(x).$$  Then it is easy to show that with $A(0,t)=0$ that $U$ satisfies the inviscid Burgers equation  

$$U_t(x,t)+U(x,t)U_x(x,t)=0 , \:\:\:\: U(x,0)=F(x).$$

For details, see my sketch "Compositional inverse pairs, the Burgers-Hopf equation, and associahedra" at my [mini-arxiv][8].

With $F(x)=c_2\:x^2+c_3\:x^3+ \cdots\;$, we have as asserted in Example I that

$$A(x,t)=x+(-c_2t)x^2+(-c_3t+2c_2^2t^2)x^3+(-c_4t+5c_2c_3t^2-5c_2^3t^3)x^4+(-c_5t+(6c_2c_4+3c_3^2)t^2+21c_2^2c_3t^3+14c_2^4t^4)x^5+\cdots\:,$$

the associahedra again. For $F(x)=x^n$, with $n>2$, $A(x,t)$ is the o.g.f. for the Fuss-Catalan numbers, which are related to dissections of polygons (cf. OEIS-[A001764][9], particularly the Schuetz/Whieldon link). For $n=2$, we obtain the celebrated Catalan numbers and relations to Brownian motion, Lax pairs, random matrix theory, and Wigner's semicircle law/distribution, as discussed by Govind Meno in "[Burgers turbulence: kinetic theory and complete integrability][10]" and a similarly titled paper by Ravi [Srinivasan][11]. Victor Buchstaber in "[Toric Topology of Stasheff Polytopes][12]" even derives the Catalan numbers from an infinite set of conservation laws reminiscent of those for the KdV equation.


  [1]: http://www.scholarpedia.org/article/Dynamical_systems
  [2]: http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines
  [3]: http://en.wikipedia.org/wiki/Integral_curve
  [4]: http://www2.maths.ox.ac.uk/cmi/library/academy/LectureNotes05/Lodaypaper.pdf
  [5]: https://oeis.org/A133437
  [6]: http://arxiv.org/abs/0910.0120
  [7]: http://en.wikipedia.org/wiki/Burgers%27_equation
  [8]: http://tcjpn.wordpress.com/
  [9]: https://oeis.org/A001764
  [10]: https://www.icts.res.in/media/uploads/Old_Talks_Lectures/Document/1265006269Govind%20Menon.pdf
  [11]: https://www.ma.utexas.edu/users/rav/docs/Srinivasan.ESI.pdf
  [12]: http://eprints.ma.man.ac.uk/998/01/covered/MIMS_ep2007_232.pdf