I am trying to understand how to construct the Bruhat-Tits building of $SL_2(\mathbb{Q}_p)$. I am reading the general construction of the semisimple Bruhat-Tits building of a connected reductive group. I will explain what I think is the construction (but please correct me if anything I'm saying is wrong) of the building as a quotient of $SL_2 \times A$ by an equivalence relation.
Let $T$ be the maximal torus of diagonal matrices. The apartment $A$ of $SL_2$ associated to $T$ is the tensor product of the cocharacters of $T$ with $\mathbb{R}$: it can be identified with $\mathbb{R}$. For $x \in \mathbb{R}$ let $U_x$ be the subgroup of $SL_2(\mathbb{Q}_p)$ generated by the identity, the matrices with 1's on the diagonal, 0 on the lower left, and $c \in \mathbb{Q_p}$ such that val$(c) \geq -2x$, and the matrices with 1's on the diagonal, 0 on the top right, and $d$ with val$(d) \geq 2x$. Let $N$ be the normalizer of $T$ (so the monomial matices with entries in $\mathbb{Q}_p$). It acts on $A$:
$\left( \begin{array}{cc} t & 0 \\ 0 & t^{-1} \\ \end{array} \right)$ sends $x$ to $x-$val$(t)$ and
$\left( \begin{array}{cc} 0 & t\\ -t^{-1} & 0 \\ \end{array} \right)$ sends $x$ to $-x-$val$(t)$.
Define an equivalence relation $\sim$ on $SL_2(\mathbb{Q}_p) \times A$ by:
$(g,x) \sim (h,y)$ if there is $n \in N$ such that $nx=y$ and $g^{-1}hn \in U_x$. The quotient is the building $X$ of $SL_2(\mathbb{Q}_p)$.
The apartment $A$ can be identified as a subset of $X$ by $x \mapsto$ the class of $(Id, x)$.
Somehow this is a 3-regular tree. This is what I don't understand. The extended building of $GL_2$ can be thought of as the set of additive norms on $(\mathbb{Q}_p)^2$. If $T_1$ is the diagonal torus of GL$_2$, the apartment can be identified with $\mathbb{R}^2$ and the elements of the apartment can be identified with additive norms by sending $(a_1,a_2)$ to the norm $\phi$ defined by $\phi(x_1,x_2)=$inf$\{val(x_1)-a_1,val(x_2)-a_2\}$.
$\mathbb{R}$ acts on these norms by translating the values of the norm. We can identify the apartment of $SL_2$ with norms-on-$\mathbb{Q}_p^2$-up-to-translation: $a \in \mathbb{R}$ would correspond to the equivalence class of the norm $\phi(x_1,x_2)=$inf$\{val(x_1),val(x_2)-a\}$.
$SL_2$ acts on the building $X$: $g$ sends the class of $(h,x)$ to the class of $(gh,x)$. This corresponds to an action of the norms: $g$ sends $\phi$ to the norm $g\phi$ defined by $g\phi(v)=\phi(gv)$ where $v \in \mathbb{Q}_p^2$.
If all this is true, then two equivalent pairs $(g,x)$ and $(h,y)$ of $SL_2 \times A$ should give the same norm-up-to-translation. But I don't think this is true.
My question is: How does one go from the abstract construction of the building to get a tree? I haven't found any literature that explains this.