A finite graph embedded in the plane is called a series-parallel map if it can be obtained from a finite tree by repeatedly subdividing and doubling edges. We study the large-scale geometry of random two-connected series-parallel maps with n edges. Our main result shows that, when graph distances are rescaled by a factor n^{-1/2}, these maps converge to a constant multiple of Aldous' continuum random tree.
The proof relies on a bijection between series-parallel maps with n edges and a class of trees with n leaves. This correspondence allows us to compare geodesics in the maps with paths in the associated trees which enables us to transfer metric information from the trees to the maps.