It is well known since M.Hata's 1985 paper that If a self-similar set has a connected intersection graph, then it is connected and locally connected. On the other hand, there are self-similar sets with a finite number of simply connected and locally connected components such as self-similar tiles. We proved that if a fractal square or a Bedford McMullen carpet has a finite number of simply-connected components then these components are dendrites and the attractor itself is a fractal forest, which contains a finite connection of main trees, which defines the structure of this fractal forest.
