Distinguishing locally finite trees

The distinguishing number \(D(G)\) of a graph \(G\) is the smallest number of colors that is needed to color the vertices of \(G\) such that the only color preserving automorphism is the identity. For infinite graphs \(D(G)\) is bounded by the supremum of the valences, and for finite graphs by \(\Delta(G)+1\), where \(\Delta(G)\) is the maximum valence. Given a finite or infinite tree \(T\) of bounded finite valence \(k\) and an integer \(c\), where \(2 \leq c \leq k\), we are interested in coloring the vertices of \(T\) by \(c\) colors, such that every color preserving automorphism fixes as many vertices as possible. In this sense we show that there always exists a \(c\)-coloring for which all vertices whose distance from the next leaf is at least \(\lceil\log_ck\rceil\) are fixed by any color preserving automorphism, and that one can do much better in many cases.