$\DeclareMathOperator\SL{SL}\DeclareMathOperator\tr{\mathsf{tr}}$$\SL_2(\mathbb{N})$ is a free monoid on the generators
$$
L=\begin{pmatrix}1&0\\1&1\end{pmatrix},\quad R=\begin{pmatrix}1&1\\0&1\\\end{pmatrix}.
$$
Let $\SL_2^{(n)}(\mathbb{N})$ designate the set of words of length $n$ in these generators.  I have a couple of questions about this family that seem within reach (i.e. are already known or could be determined by the good folks of mathoverflow).

1. What is the distribution of $\{\tr(M) : M\in \SL_2^{(n)}(\mathbb{N})\}$?
2. What is the distribution of $\{\|M\|_{\infty} : M\in \SL_2^{(n)}(\mathbb{N})\}$?
3. What is the joint distribution of the two quantities above?

I'd prefer to know the limiting distribution $n\to\infty$ and some useful error term (if they exist), but any non-trivial bounds are welcome, or approximations to relevant functions of the matrices as functions of the $\{L,R\}$ decomposition
$$
M(w)=\prod_{i=0}^{n-1}G_{w_i}, \quad w\in\{0,1\}^n, \quad G_0=L, \quad G_1=R.
$$

Here are a few pictures (of dubious quality); scatterplot on top of bar plot (could've just done a histogram but was more curious about some of the exact values).

Traces, $n=15,20,25$:
[![trace counts for length 15 words][1]][1]
[![trace counts for length 20 words][2]][2]
[![trace counts for length 25 words][3]][3]

$L^{\infty}$, $n=15,20,25$:
[![sup norm counts for length 15 words][4]][4]
[![sup norm counts for length 20 words][5]][5]
[![sup norm counts for length 25 words][6]][6]

Joint $n=15$ (the linear relationship seems to be $\tr(M)\approx \frac{9}{8}\|M\|_{\infty}$):
[![Joint n = 15][7]][7]


  [1]: https://i.sstatic.net/7ATODAfe.png
  [2]: https://i.sstatic.net/2fyUumCM.png
  [3]: https://i.sstatic.net/kEDTsZDb.png
  [4]: https://i.sstatic.net/8M9IVfET.png
  [5]: https://i.sstatic.net/WxAX6raw.png
  [6]: https://i.sstatic.net/tCiw9oLy.png
  [7]: https://i.sstatic.net/33DWqOlD.png