The structure of molecular clouds holds important clues regarding the physical processes that lead to their formation and subsequent dynamical evolution. While it is well established that turbulence imprints a self-similar structure onto the clouds, other processes, such as gravity and stellar feedback, can break their scale-free nature. The break of self-similarity can manifest itself in the existence of characteristic scales that stand out from the underlying structure generated by turbulent motions. In this work, we investigate the structure of the Cygnus-X North and Polaris Flare molecular clouds, which represent two extremes in terms of their star formation activity. We characterize the structure of the clouds using the delta-variance (Δ-variance) spectrum. In the Polaris Flare, the structure of the cloud is self-similar over more than one order of magnitude in spatial scales. In contrast, the Δ-variance spectrum of Cygnus-X North exhibits an excess and a plateau on physical scales of ≈0.5−1.2 pc. In order to explain the observations for Cygnus-X North, we use synthetic maps where we overlay populations of discrete structures on top of a fractal Brownian motion (fBm) image. The properties of these structures, such as their major axis sizes, aspect ratios, and column density contrasts with the fBm image, are randomly drawn from parameterized distribution functions. We are able to show that, under plausible assumptions, it is possible to reproduce a Δ-variance spectrum that resembles that of the Cygnus-X North region. We also use a “reverse engineering” approach in which we extract the compact structures in the Cygnus-X North cloud and reinject them onto an fBm map. Using this approach, the calculated Δ-variance spectrum deviates from the observations and is an indication that the range of characteristic scales (≈0.5−1.2 pc) observed in Cygnus-X North is not only due to the existence of compact sources, but is a signature of the whole population of structures that exist in the cloud, including more extended and elongated structures.