  We study the dynamical behaviour of weighted backward shift operators defined
on sequence spaces over a directed tree. We provide a characterization of chaos
on very general Fr\'echet sequence spaces in terms of the existence of a large
supply of periodic points, or of fixed points. In the special case of the space
$\ell^p$, $1\leq p<\infty$, or the space $c_0$ over the tree, we provide a
characterization directly in terms of the weights of the shift operators. It
has turned out that these characterizations involve certain generalized
continued fractions that are introduced in this paper. Special attention is
given to weighted backward shifts with symmetric weights, in particular to
Rolewicz operators. In an appendix, we complement our previous work by
characterizing hypercyclic and mixing weighted backward shifts on very general
Fr\'echet sequence spaces over a tree. Also, some of our results have a close
link with potential theory on flows over trees; the link is provided by the
notion of capacity, as we explain in an epilogue.
