As you can see, I have been studying analytic combinatorics, specifically Flajolet's symbolic method. And while I was reading about how combinatorial structures are translated into generating functions, the question arose as to whether this is a bijective or double counting proof. To contextualize I will give an example:
To count the number of binary trees with a fixed number of internal nodes, a symbolic expression of this type is obtained $\mathcal{T}=\{\square\}\cup(\{\bullet\}\times\mathcal{T}\times\mathcal{T})$, and assuming that $\mathcal{T}$ has the generating function $T(z)$ then this translates to $1+ z\cdot T(z)\cdot T(z)= 1+zT(z)²$. From this expression we obtain that
$\displaystyle T(z)=\frac{1-\sqrt{1-4z}}{2z}$
and
$\displaystyle [z^n]T_n=\frac{1}{n+1}\binom{2n}{n}$,
and thus the number of flat binary trees with $n$ internal nodes is given by the nth Catalan number.
Does this type of reasoning correspond to a bijective or double counting proof? I can't get the answer clear.