Aims. For many years feedback processes generated by OB-stars in molecular clouds, including expanding ionization fronts, stellar winds, or UV-radiation, have been proposed to trigger subsequent star formation. However, hydrodynamic models including radiation and gravity show that UV-illumination has little or no impact on the global dynamical evolution of the cloud. Instead, gravitational collapse of filaments and/or merging of filamentary structures can lead to building up dense high-mass star-forming clumps. However, the overall density structure of the cloud has a large influence on this process, and requires a better understanding. Methods. The Rosette molecular cloud, irradiated by the NGC 2244 cluster, is a template region for triggered star-formation, and we investigated its spatial and density structure by applying a curvelet analysis, a filament-tracing algorithm (DisPerSE), and probability density functions (PDFs) on Herschel column density maps, obtained within the HOBYS key program. Results. The analysis reveals not only the filamentary structure of the cloud but also that all known infrared clusters except one lie at junctions of filaments, as predicted by turbulence simulations. The PDFs of sub-regions in the cloud show systematic differences. The two UV-exposed regions have a double-peaked PDF we interprete as caused by shock compression, while the PDFs of the center and other cloud parts are more complex, partly with a power-law tail. A deviation of the log-normal PDF form occurs at A(V) approximate to 9(m) for the center, and around 4(m) for the other regions. Only the part of the cloud farthest from the Rosette nebula shows a log-normal PDF. Conclusions. The deviations of the PDF from the log-normal shape typically associated with low-and high-mass star-forming regions at A(V) approximate to 3-4(m) and 8-10(m), respectively, are found here within the very same cloud. This shows that there is no fundamental difference in the density structure of low-and high-mass star-forming regions. We conclude that star-formation in Rosette - and probably in high-mass star-forming clouds in general - is not globally triggered by the impact of UV-radiation. Moreover, star formation takes place in filaments that arose from the primordial turbulent structure built up during the formation of the cloud. Clusters form at filament mergers, but star formation can be locally induced in the direct interaction zone between an expanding HII-region and the molecular cloud.