Variable temperature scanning tunneling microscopy at substrate temperatures between 25 and 550 K reveals a transition from diffusion limited aggregation to thermodynamic equilibrium in the two-dimensional (2D) monolayer growth of Ag on the Pt(111) surface. Via deposition temperature control the ''nanostructure'' of the grown 2D-Ag-films can be tuned in a very defined way: from randomly distributed adatoms and small clusters, through highly dendritic islands, dense large islands with irregular or smooth boundaries to an equilibrium 2D condensate-gas mixture.