  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,
chi_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 in 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) is in L(v) for all v. The list distinguishing number of G, D_l(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 chi_{D_l}(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_l(T) = D(T), chi_D(T)=chi_{D_l}(T), and chi_D(T) <= D(T) + 1.
