List distinguishing parameters of trees

A coloring of the vertices of a graph G is said to be distinguishing provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, @g"D(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring. Recently the notion of a distinguishing coloring was extended to that of a list distinguishing coloring. Given an assignment L={L(v)}"v"@?"V"("G") of lists of available colors to the vertices of G, we say that G is (properly) L-distinguishable if there is a (proper) distinguishing coloring f of G such that f(v)@?L(v) for all v. The list distinguishing number of G, D"@?(G), is the minimum integer k such that G is L-distinguishable for any list assignment L with |L(v)|=k for all v. Similarly, the list distinguishing chromatic number of G, denoted @g"D"""@?(G) is the minimum integer k such that G is properly L-distinguishable for any list assignment L with |L(v)|=k for all v. In this paper, we study these distinguishing parameters for trees, and in particular extend an enumerative technique of Cheng to show that for any tree T, D"@?(T)=D(T), @g"D(T)[email protected]"D"""@?(T), and @g"D(T)@?D(T)+1.