List distinguishing parameters of trees

A coloring of the vertices of a graph GG is said to be distinguishing   provided no nontrivial automorphism of GG preserves all of the vertex colors. The distinguishing number   of GG, D(G)D(G), is the minimum number of colors in a distinguishing coloring of GG. The distinguishing chromatic number   of GG, χD(G)χD(G), is the minimum number of colors in a distinguishing coloring of GG 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)L={L(v)}v∈V(G) of lists of available colors to the vertices of GG, we say that GG is (properly) LL-distinguishable   if there is a (proper) distinguishing coloring ff of GG such that f(v)∈L(v)f(v)∈L(v) for all vv. The list distinguishing number   of GG, Dℓ(G)Dℓ(G), is the minimum integer kk such that GG is LL-distinguishable for any list assignment LL with |L(v)|=k|L(v)|=k for all vv. Similarly, the list distinguishing chromatic number   of GG, denoted χDℓ(G)χDℓ(G) is the minimum integer kk such that GG is properly LL-distinguishable for any list assignment LL with |L(v)|=k|L(v)|=k for all vv.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 TT, Dℓ(T)=D(T)Dℓ(T)=D(T), χD(T)=χDℓ(T)χD(T)=χDℓ(T), and χD(T)≤D(T)+1χD(T)≤D(T)+1.