  Inspired by a recent work of Crovisier and Pujals on mildly dissipative
diffeomorphisms of the plane, we show that H\'enon-like and Lozi-like maps on
their strange attractors are conjugate to natural extensions (a.k.a. shift
homeomorphisms on inverse limits) of maps on metric trees with dense set of
branch points. In consequence, these trees very well approximate the topology
of the attractors, and the maps on them give good models of the dynamics. To
the best of our knowledge, these are the first examples of canonical
two-parameter families of attractors in the plane for which one is guaranteed
such a 1-dimensional locally connected model tying together topology and
dynamics of these attractors. For the H\'enon maps this applies to a positive
Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters,
the Wang-Young parameter set, and sheds more light onto the result of Barge
from 1987, who showed that there exist parameter values for which H\'enon maps
on their attractors are not natural extensions of any maps on branched
1-manifolds. For the Lozi maps the result applies to an open set of parameters
given by Misiurewicz in 1980. Our result can be seen as a generalization to the
non-uniformly hyperbolic world of a classical result of Williams from 1967. We
also show that no simpler 1-dimensional models exist.
