In this joint work with Sonja Štimac, we extend R.F. Williams’ result on 1-dimensional hyperbolic attractors to the non-uniformly hyperbolic setting, by showing that each Wang-Young strange attractor in the plane is conjugate to the shift on the inverse limit of a baobab (Peano continuum that contains at most one Jordan curve), generalizing our earlier result on Hénon attractors. More generally, the result holds on the core of the maximal attractor of any mildly dissipative diffeomorphism (in the sense of Crovisier and Pujals). We also generalize these results to the C⁰ setting, by introducing the class of topologically mildly dissipative surface homeomorphisms, providing a unified approach that covers many classes of dissipative dynamical systems scattered in the literature. Our purely topological conditions lead to a one-to-one correlation between the sets of ergodic measures of the one-dimensional and two-dimensional systems, as well as equality between the corresponding measure-theoretic entropies.