  In this paper we investigate a new class of operators called weighted shifts
on directed trees introduced recently in [Z. J. Jablonski, I. B. Jung and J.
Stochel, A Non-hyponormal Operator Generating Stieltjes Moment Sequences, J.
Funct. Anal. 262 (2012), no. 9, 3946--3980.]. This class is a natural
generalization of the so called weighted bilateral, unilateral and backward
shift operators. In the first part of the paper we calculate the asymptotic
limit and the isometric asymptote of a contractive weighted shift on a directed
tree and that of the adjoint. Then we use the asymptotic behaviour and
similarity properties to deal with cyclicity. We also show that a weighted
backward shift operator is cyclic if and only if there is at most one zero
weight.
