Two smooth tangent almost complex curves in a $4$-manifold I would like to know if following is correct.
Statement. Suppose we have a smooth (i.e., $C^\infty$) almost  complex structure on $\mathbb R^4$ and $C_1, C_2$ are two $J$-holomorphic curves passing through $(0,0)$, tangent at $(0,0)$ and regular at $(0,0)$. Then there exist $C^{\infty}$ smooth complex coordinates $(z,w)$ such that $C_1$ is locally given by $w=0$ and $C_2$ by $w=z^n$.
PS. I think it would be enough for me to know that $C_2$ can be given by $w=z^n+O|z^{n+1}|$. However everything must be $C^{\infty}$ - the coordinates, and the $O|z^{n+1}|$ term. If, on the other hand one can not expect to have a $C^{\infty}$ diffeo, what is the best one can expect?
 A: This follows from theorem 6.2 (and the first sentence in the proof) of Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2) 141 (1995), no. 1, 35–85.
A: It turns out that it is nice to read books. The answer to the weaker version of my question with $O|z|^{n+1}$ term is contained on page 17 of McDuff-Salamon book [MS] (no need of Micallef-White!):
https://people.math.ethz.ch/~salamon/PREPRINTS/jholsm.pdf
Proof. In the proof of Lemma 2.2.3 of [MS] one uses coordinates in $\mathbb C^2$ such that $C_1$ is given by $w=0$ and the almost complex structure $J$ along the line $(z,0)$ is the standard one. Then it is explained that the almost complex map $z\to \mathbb C^2$ corresponding to $C_2$ is given by 
$$z\to (p(z)+O(|z^{n+1}|), az^n+O(|z^{n+1}|))$$
where $p(z)$ is a polynomial of order at most $n$, $a\ne 0$. In our case of course $p'(0)\ne 0$. It is now clear that in these coordinates $C_2$ is as need. QED.
Comment. The above proof is elementary and does not use Micallef-White. Similarly to Micallef-White's, statement it can be used to answer the original question with a $C^1$-smooth change of coordinates (instead of $C^{\infty}$). Indeed, after a smooth reparameterization in $z$ and scaling in $w$ the above map for $C_2$ looks as
$$z\to (z, z^n+O(|z^{n+1}|)).$$
Denote the second term by $f(z)$. Then the map
$(z,w)\to (z,w-(f(z)-z^n)/z^n)$ is $C^1$ and it sends the couple $C_1,C_2$ to the couple $(w=0, w=z^n)$. 
I wonder still if one can make this last change of coordinates $C^{\infty}$...
