Similarity and asymmetrization of trees (Q686304)

An asymmetrizing set of trees \(T\) is a set \(A\) of vertices of \(T\) such that the identity is the only automorphism of \(T\) which stabilizes \(A\). The similarity number \(s(T)\) (resp. asymmetrizing number \(a(T))\) of \(T\) is the cardinality of the set of orbits of subsets of vertices (resp. asymmetrizing sets) of \(T\). The author studies these two numbers for various kinds of (finite and infinite) trees. As main result a characterization of trees \(T\) with \(a(T)=2^{| T |}\) (resp. \(s(T)=2^{| T |})\) is given.