Similarity and asymmetrization of trees

AbstractAn asymmetrizing set of a tree T is a set A of vertices of T such that the identity is the only automorphism of T which stabilizes A. The similarity (resp. asymmetrizing) number of T is the cardinality of the set of orbits of subsets of vertices (resp. asymmetrizing sets) of T. We study these two numbers for various kinds of trees; and, extending a result of [2] about trees without endpoints, we characterize the trees T which have, as well as other nice properties, an asymmetrizing (resp. similarity) number equal to 2|T|