I’m studying Chapter II: $\mathbf{SL}_2$ of Serre’s book “Trees”. In paragraph 1, Serre defines the tree of $SL_2$ over a local field $K$. In particular, he considers the set of $\mathcal O$-lattices of $K^2$ (where $\mathcal O=\{x\in K\vert\ \nu(x)\geq 0\}$ is the ring of integers of $(K,\nu)$) modulo the action of $K^{\times}:=K\setminus\{0\}$ by left multiplication. The set of lattice classes such is the set of vertices of a combinatorial graph $X$. In $\mathbf{Theorem\ 1}$, he proves that $X$ is a tree: connectedness is clear, while the fact that $X$ simply connected is not well explained. At some point (pag. 70) he says:
To prove that $X$ is a tree, it now suffices to show that, if $\Lambda_0,\dots,\Lambda _n$, $n\geq 1$ is the sequence of vertices in a path without backtracking m in $X$, then $\Lambda_0\neq \Lambda _n$. In fact we shall show (by induction on n) that $d(\Lambda_0,\Lambda _n)=n$. From what we have said above, we can find representatives $L_i\in \Lambda _i$ such that $L_{i+1}\subset L_i$ and $l(L_i/L_{i+1})=1$. We have $l(L_0/L_n)=n$ and we want to show $L_n\not\subset L_0\pi$. By induction hypothesis we have $L_{n-1}\not\subset L_0\pi$. The lattices $L_n$ and $L_{n-2}\pi$ are the inverse images of two lines in the $k=\mathcal O/\pi\mathcal O$-plane $L_{n-1}/L_{n-1}\pi$.
I don’t understand what does it mean by $\mathit{lines}$. I thought that, since $L_n\subset L_{n-1}$ and $L_{n-1}/L_{n-1}\pi\simeq k^2$, $L_n $ is canonically embedded in it; but what about $L_{n-2}\pi$?
Note. I’m assuming $ K$ to be commutative.