Distinguishing finite and infinite trees of arbitrary cardinality

Let $G$ be a finite or infinite graph and $m(G)$ the minimum number of vertices moved by the non-identity automorphisms of $G$. We are interested in bounds on the supremum $Δ(G)$ of the degrees of the vertices of $G$ that assure the existence of vertex colorings of $G$ with two colors that are preserved only by the identity automorphism, and, in particular, in the number $a(G)$ of such colorings that are mutually inequivalent. For trees $T$ with finite $m(T)$ we obtain the bound $Δ(T)\leq2^{m(T)/2}$ for the existence of such a coloring, and show that $a(T)= 2^{|T|}$ if $T$ is infinite. Similarly, we prove that $a(G) = 2^{|G|}$ for all tree-like graphs $G$ with $Δ(G)\le 2^{\aleph_0}$. For rayless or one-ended trees $T$ with arbitrarily large infinite $m(T)$, we prove directly that $a(T)= 2^{|T|}$ if $Δ(T)\le 2^{m(T)}$.