Since it is regular of countably infinite degree, it is isomorphic to the union of Bruhat-Tits trees for PGL₂(F) as F ranges over unramified extensions of a local field. The automorphism group therefore contains PGL₂(F^nr), but I don't know much more.

Edit: The automorphism group is apparently described in a 1970 paper by Tits: Sur le groupe des automorphismes d'un arbre, which is not available on line. There is a description of its subgroups of cardinality less than continuum in Moller's paper The automorphism groups of regular trees - any such subgroup is either the simple index 2 subgroup generated by stabilizers of points, or lies between the pointwise stabilizer of a finite (possibly empty) subtree and the setwise stabilizer of the same subtree. The stabilizer of a point is a limit of a system of wreath products of Sym(Aleph₀).

Since it is regular of countably infinite degree, it is isomorphic to the union of Bruhat-Tits trees for PGL₂(F) as F ranges over unramified extensions of a local field. The automorphism group therefore contains PGL₂(F^nr), but I don't know much more.

Edit: The automorphism group is apparently described in a 1970 paper by Tits: Sur le groupe des automorphismes d'un arbre, which is not available on line. There is a description of its subgroups of index less than continuum cardinality in Moller's paper The automorphism groups of regular trees - any such subgroup is either the simple index 2 subgroup generated by stabilizers of points, or lies between the pointwise stabilizer of a finite (possibly empty) subtree and the setwise stabilizer of the same subtree. The stabilizer of a point is a limit of a system of wreath products of Sym(Aleph₀).

Since it is regular of countably infinite degree, it is isomorphic to the union of Bruhat-Tits trees for PGL₂(F) as F ranges over unramified extensions of a local field. The automorphism group therefore contains PGL₂(F^nr), but I don't know much more.

Since it is regular of countably infinite degree, it is isomorphic to the union of Bruhat-Tits trees for PGL₂(F) as F ranges over unramified extensions of a local field. The automorphism group therefore contains PGL₂(F^nr), but I don't know much more.

Edit: The automorphism group is apparently described in a 1970 paper by Tits: Sur le groupe des automorphismes d'un arbre, which is not available on line. There is a description of its subgroups of cardinality less than continuum in Moller's paper The automorphism groups of regular trees - any such subgroup is either the simple index 2 subgroup generated by stabilizers of points, or lies between the pointwise stabilizer of a finite (possibly empty) subtree and the setwise stabilizer of the same subtree. The stabilizer of a point is a limit of a system of wreath products of Sym(Aleph₀).