Asymmetrizing infinite trees

A graph $G$ is asymmetrizable if it has a set of vertices whose setwise stablizer only consists of the identity automorphism. The motion $m$ of a graph is the minimum number of vertices moved by any non-identity automorphism. It is known that infinite trees $T$ with motion $m=\aleph_0$ are asymmetrizable if the vertex-degrees are bounded by $2^m.$ We show that this also holds for arbitrary, infinite $m$, and that the number of inequivalent asymmetrizing sets is $2^{|T|}$.