In order to study the fundamental conduction mechanism of LaAlO3/SrTiO3 (LAO/STO) interfaces, heterostructures were modified with a single unit cell interface layer of either an isovalent titanate ATiO3 (A = Ca, Sr, Sn, Ba) or a rare earth modified Sr0.5RE0.5TiO3 (RE = La, Nd, Sm, Dy) between the LAO and the STO. A strong coupling between the lattice strain induced in the LAO layer by the interfacial layers and the sheet carrier density in the STO substrate is observed. The observed crystal distortion of the LAO is large and it is suggested that it couples into the sub-surface STO, causing oxygen octahedral rotation and deformation. We propose that the 'structural reconstruction' which occurs in the STO surface as a result of the stress in the LAO is the enabling trigger for two-dimensional conduction at the LAO/STO interface by locally changing the band structure and releasing trapped carriers.