Experiments of the recrystallization processes in two-dimensional complex plasmas are analyzed in order to rigorously test a recently developed scale-free phase transition theory. The Fractal-Domain-Structure (FDS) theory is based on the kinetic theory of Frenkel. It assumes the formation of homogeneous domains, separated by defect lines, during crystallization and a fractal relationship between domain area and boundary length. For the defect number fraction and system energy a scale free power-law relation is predicted. The long range scaling behavior of the bond order correlation function shows clearly that the complex plasma phase transitions are not of KTHNY type. Previous preliminary results obtained by counting the number of dislocations and applying a bond order metric for structural analysis are reproduced. These findings are supplemented by extending the use of the bond order metric to measure the defect number fraction and furthermore applying state-of-the-art analysis methods, allowing a systematic testing of the FDS theory with unprecedented scrutiny: A morphological analysis of lattice structure is performed via Minkowski tensor methods. Minkowski tensors form a complete family of additive, motion covariant and continuous morphological measures that are sensitive to non-linear properties. The FDS theory is rigorously confirmed and predictions of the theory are reproduced extremely well. The predicted scale-free power law relation between defect fraction number and system energy is verified for one more order of magnitude at high energies compared to the inherently discontinuous bond order metric. Minkowski Tensor analysis turns out to be a powerful tool for investigations of crystallization processes. It is capable to reveal non-linear local topological properties, however, still provides easily interpretable results founded on a solid mathematical framework.
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