  This paper explores the notions of $\mathcal{F}$-transitivity and topological
$\mathcal{F}$-recurrence for backward shift operators on weighted
$\ell^p$-spaces and $c_0$-spaces on directed trees, where $\mathcal{F}$
represents a Furstenberg family of subsets of $\mathbb{N}_0$. In particular, we
establish the equivalence between recurrence and hypercyclicity of these
operators on unrooted directed trees. For rooted directed trees, a backward
shift operator is hypercyclic if and only if it possesses an orbit of a bounded
subset that is weakly dense.
