The two-dimensional strain distribution in a GaN/AlN quantum-dot (QD) superlattice is measured from high-resolution transmission electron microscopy images using the geometrical phase analysis. The results are compared to elastic theoretical calculations using a combination of Fourier transform and Green's function techniques. The GaN/AlN system appears to be a model system for a comparison between theory and experiments as interdiffusion between GaN and AlN is negligible. We verify that for the case of a three-dimensional system, such as a QD, the biaxial strain approximation is not valid. Furthermore, we demonstrate that the presence of QDs induces a modulation in the strain state of the AlN barriers which is the driving force for the vertical alignment of the GaN QDs in the AlN matrix.