Higher dimensional berkovich spaces I am looking for a geometric and topological way to make a visualization of higher dimensional berkovich spaces, starting with the berkovich plane. Of course, this is just a collection of bounded semi-norms, but the question remains: 

Is there a visualization possible for $\mathbb A^2_{\text{Berk}}$ like the infinite branched tree for $\mathbb A^1_{\text{Berk}}$? 

(For $\mathbb A^1_{\text{Berk}}$ see for example Baker and Rumely's Potential Theory and Dynamics on the Berkovich projective line, Chapters 1 - 2.)
I think you get a simplicial complex, but I don't know exactly how. On the one hand (reading Favre and Johnsson's The valuative tree), you have this list of valuations (thus, also of seminorms, allthough this book discusses seminorms on $\mathbb{C}^2$). On the other hand we have Berkovich's theory of Type I - Type IV points. I guess there just more Type I - Type IV points in a plane (i.e. more seminorms that occur as it were type I points), and some can be only represented by faces (two dimensional simplices), only I don't know how.
Are there any references on the visualization part?
 A: Since I don't seem to be able to edit my own question. I found s link which might be useful here:


*

*http://users.math.yale.edu/~sp547/pdf/Anayltification-tropicalizations.pdf, in which a homeomorphism is constructed between $X^{an}$ (or $X_{Berk}$, if that is you cup of coffee) and an inverse limit of tropicalizations of embeddings of (toric) subvarieties of $X(K)$, where $K$ is the basefiel. Payne first takes the field $\mathbb{C}((t^{\mathbb{R}}))$ (in which, for example in $\mathbb{P}^1_{Berk,K}$ there are only Type 1 and Type 2 points), then switches to fields with trivial norms.

A: In fact you get more type of points, I believe. Points of type I are just k-point, i.e. points such that H(x)=k, you have a general fact that if l is algebraic over the completion l' of trascendence degree n over k then the sum of the trancendence degree of the reduction of l over the reduction of k and the rank of |l*| over |k*| is less or equal than n.
In the projective line, as in all the 1-dimentionals analytics spaces, this implies that you have only four types of points (H(x)=k, OR |H(x)*| increase of one degree of trascendence, OR the reduction of H(x) increase of "one rank", OR H(x) it is a non trivial immediate extention of k). When n>1 you may have then a lot more types of points, and the number of types increase when n grows.
If you want to find those generals informations there's a paper of Temkin wich is very intresting.
A: @David. The best reference I was able to find (of some relevance) is "The Valuative Tree" by Matthias Johnsson, which discusses norms on $\mathbb{C}[x,y]$. The only thing which isn't clear from this is, which points are Type I, ..., Typer IV. 
You don't get more types (so no Type V or something like that) [this is proven somewhere in Berkovich's Analysis on Non-archimedean spaces], but I think you get more Type I, more Type II, etc points.
