I understand that $SL(2,\Bbb{Q}_p)$ acts on an infinite (p+1)-regular tree, and that the subgroup $SL(2,\Bbb{Z}_p)$ acts by fixing a vertex, rotating the (p+1) neighbors by the $SL(2,\Bbb{Z}/p)$-action, the neighbors of those neighbors by the $SL(2,\Bbb{Z}/p^2)$-action, etc.
How does the matrix $$ \left[ \begin{array}{rr} 1 & 1/p \\ 0 & 1 \end{array} \right] $$ act on the tree? I keep getting confused.