TITLE: What is the "serious" name for the topograph (for a quadratic form)
QUESTION [20 upvotes]: One way to study (mixed signature) quadratic forms in two variables is to study the topograph.  Looks like the signature doesn't matter:  here is (1,1) and (1-,1).
The name is derived from τοποσ (Greek:  "place") and γραφή ("writing").  I read that if you're really good at reading topographs you can extract information like the genus, class number, solve the Pell equation, and more.

There are two resources I found for topographs:

The Sensual Quadratic Form, John H Conway
Topology of Numbers, Allen Hatcher

They are basically drawing the dual tree of the Farey Tesselation, which is a tiling of $\mathbb{H}$ or $\mathbb{D}$ by hyperbolic triangles.  Is there a more serious name for putting trees on $\mathbb{H}$?
This question emerges, for example, trying to draw these things with a computer and I needed to decide a natural place to put the interior vertices, and I couldn't think of one.  The  "outer" vertices are indexed by $\text{P}\mathbb{Q}^1$ and the interior vertices could be in any reasonable place.
There could be a serious name for this structure, like the Bruhat-Tits building or maybe it's in Serre's book on Trees.  Any guidance?


A figure similar to the topograph also appears in a discussion of the Bruhat-Tits tree for $\text{SL}(2, \mathbb{Q}_2)$. [notes]

REPLY [4 votes]: There is a somewhat larger machinery - a generalization of the theory of dessins d'enfants, when drawing the topograph of a binary quadratic form. 
The topograph is merely the quotient of the bipartite Farey tree by the translation action of the correct subgroup of the modular group--where correct means the automorphism group of the binary quadratic form in question. The quotient looks like a wheel so we call it a çark (pronounced as chark, the word has a common etymology with Indian chakra, Greek kyklos and English wheel). 
Using the çark one can solve many problems around the corresponding binary quadratic form. Details can be found here and here. 
There is also an android application here.