TITLE: Series defined by a fixed-point functional equation
QUESTION [12 upvotes]: Description
I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here is a description of the objects.
Let $X := \lbrace x_1, \dots, x_k \rbrace$ be an alphabet of mutually commuting indeterminates and $F(x_1, \dots, x_k)$ be the formal power series
\begin{equation}
    F(x_1, \dots, x_k) := 
    \sum_{\alpha_1, \dots, \alpha_k} 
    \lambda_{\alpha_1 \dots \alpha_k} \: 
    x_1^{\alpha_1} \dots x_k^{\alpha_k},
\end{equation}
where all coefficients $\lambda_{\alpha_1 \dots \alpha_k}$ are nonnegative integers.
The series $F(x_1, \dots, x_k)$ is defined by the fixed-point functional equation
\begin{equation}
    F(x_1, \dots, x_k) = x_1 + F(P_1(x_1, \dots, x_k), \dots, P_k(x_1, \dots, x_k)),
\end{equation}
where for any $i \in [k]$, $P_i(x_1, \dots, x_k)$ is a polynomial with nonnegative integer coefficients. Of course, there are some necessarily conditions to ensure that $F(x_1, \dots, x_k)$ is well-defined.
Here are two celebrated examples of fixed-point equations of this kind.

The series $S(x)$ defined by
\begin{equation}
    S(x) = x + S(x^2 + x^3).
\end{equation}
This functional equation has been studied by A. M. Odlyzko (see [Odl82]). One can compute first coefficients of $S(x)$ by iteration:
\begin{equation}
    S_1(x) = x,
\end{equation}
\begin{equation}
    S_2(x) = x + x^2 + x^3,
\end{equation}
\begin{equation}
    S_3(x) = x + x^2 + x^3 + x^4 + 2x^5 + 3x^7 + 3x^8 + x^9,
\end{equation}
and so on. This generating series is of the form
\begin{equation}
    S(x) = x + x^2 + x^3 + x^4 + 2x^5 + 2x^6 + 3x^7 + 4x^8 + 5x^9 + 
    8x^{10} + 14x^{11} + 23x^{12} + \cdots
\end{equation}
and its coefficients have a combinatorial interpretation: the coefficient of $x^n$ is the number of $2,3$-balanced trees (see [Odl82] and also A014535) with $n$ leaves. 
The series $B(x, y)$ defined by
\begin{equation}
    B(x, y) = x + B(x^2 + 2xy, x).
\end{equation}
By iteration, one finds
\begin{equation}
    B_1(x, y) = x,
\end{equation}
\begin{equation}
    B_2(x, y) = x + 2xy + x^2,
\end{equation}
\begin{equation}
    B_3(x, y) = x + 2xy + x^2 + 4x^2y + 2x^3 + 4x^2y^2 + 4x^3y + x^4.
\end{equation}
The coefficient of $x^n$ in the specialization $B(x) := B(x, 0)$ is the number of balanced binary trees (see [BLL94] and also A006265) with $n$ leaves.

In my work, I encountered fixed-point functional equations rather more complicated:
\begin{equation}
    F(x, y, z) = x + F(x^2 + xy + yz, x, xy),
\end{equation}
\begin{equation}
    G(x, y, z) = x + G(x^2 + 2xy + yz, x, x^2 + xy),
\end{equation}
\begin{equation}
    H(x, y, z, t) = x + H(x^2 + 2yt + yz, x, x^2 + xy, yt + yz).
\end{equation}
Coefficients of these count some tree-like structures.
It seems that the theory is fairly well-known when the generating series are univariate (for instance, A. M. Odlyzko [Odl82] gives asymptotic formulas and recurrence relations for the coefficients of $S(x)$). In the multivariate case, it seems that only few things are known and up to my knowledge, there are no other way to express the series $B(x)$ counting balanced binary trees.
Questions

(1) Have you encountered other fixed-point functional equations of the described form?

Given a fixed-point functional equation for a series $F$ of the given form:

(2) What about an other method to extract coefficients of $F$ otherwise than iteration?
(3) What about recurrence relations between the coefficients of $F$?
(4) What about asymptotic approximations for the coefficients of $F$?
(5) What about rationality, algebraicness, d-finiteness, or other of $F$?

References
[Odl82] A. M. Odlyzko. Periodic Oscillations of Coefficients of Power Series That Satisfy Functional Equations. Advances in Mathematics, 44:180–205, 1982
[BLL94] F. Bergeron, G. Labelle, and P. Leroux. Combinatorial Species and Tree-like Structures. Cambridge University Press, 1994.

REPLY [2 votes]: This maybe off-topic answer, but since OP asked 
(1) Have you encountered other fixed-point functional equations of the described form?
I will put an example that I have seen. 
$$
S(x)=x-S(x^2)
$$
This gives the power series
$$
S(x)=\sum_{k=0}^{\infty} (-1)^k x^{2^k}.
$$
This is the famous example given by Hardy. 
Using Tauberian Theory, one can prove that this is not Abel summable. i.e.
$$
\lim_{x\rightarrow 1-} S(x) \textrm{ does not exist.}
$$
cf) G. H. Hardy, “On certain oscillating series”, Quart. J. Math. 38 (1907), 269–288.