We study the internal spatial structure of 16 open clusters in the Milky Way spanning a wide range of ages. For this, we use the minimum-spanning-tree method (the Q parameter, which enables one to classify the stellar distribution as either radially or fractally clustered), King-profile fitting, and the correlation dimension (Dc) for those clusters with fractal patterns. On average, clusters with fractal-like structure are younger than those exhibiting radial stellar-density profiles. There is a significant correlation between Q and the cluster age measured in crossing-time units. For fractal clusters, there is a significant correlation between fractal dimension and age. These results support the idea that stars in newly born clusters likely follow the fractal patterns of their parent molecular clouds, and eventually evolve towards more centrally concentrated structures. However, stellar clusters as old as ~ 100 Myr can exist that have not totally destroyed their fractal structure. Finally, we have found the intriguing result that the lowest fractal dimensions obtained for the open clusters seem to be considerably smaller than the average value measured in Galactic molecular cloud complexes.