Let $G_+$ be the index 2 subgroup of $G$, consisting of automorphisms which act on $T$ without inversions, which means that if such an automorphism preserves an edge, it fixes this edge pointwise. Equivalently, this is the subgroup of $G$ generated by (pointwise) edge-stabilizers. It follows from a paper by J.Tits in 1970 ("Sur le groupe des automorphismes d'un arbre") that the subgroup $G_+$ is simple. Therefore, this is the only proper normal nontrivial subgroup of $G$.
Edit. More precisely, $G_+$ is the maximal subgroup of $G$ whose every element acts on $T$ without inversions. This subgroup necessarily has index 2 in $G$. Equivalently, if we regard $T$ as a bipartite graph, then $G_+$ is the index 2 subgroup in $G$ preserving the parts of the vertex set.