The lattice parameters of epitaxial barium strontium titanate films with various thicknesses (from 6 to 960 nm) were measured as a function of temperature in the normal and tangential directions with respect to the film plane using x-ray diffraction. The films were grown through the layer-by-layer mechanism by rf cathode sputtering under elevated oxygen pressure. A critical film thickness (similar to 50 nm) was found to exist, below and above which the films are subjected to compressive and tensile stresses, respectively. As the temperature varies from 780 to 100 K, the films undergo two diffuse structural phase transitions of the second order over the entire thickness range. The transitions in the films under tensile stresses are likely to be transformations from the paraelectric tetragonal to aa phase and then to r phase, whereas the transitions under compressive stresses are transformations from the tetragonal paraelectric to ferroelectric c phase and then, with further decreasing temperature, to r phase.