Evaluation of wave run-up characteristics is one of the most important tasks in coastal oceanography. This knowledge is required as for planning coastal structures and protection works, as for short-term tsunami forecast and tsunami warning. The nonlinear deformation and run-up of single tsunami waves of positive polarity in the conjoined water basin, which consists of the constant depth section and a plane beach is studied numerically and analytically in the framework of the nonlinear shallow water theory. Analytically, wave propagation along the constant depth section and its run-up on a beach are considered independently without taking into account wave reflection from the toe of the bottom slope. The propagation along the bottom of constant depth is described by Riemann wave, while the wave run-up on a plane beach is calculated using rigorous analytical solutions of the nonlinear shallow water theory following the Carrier-Greenspan approach. The numerical scheme employs the finite volume method and is based on the second order UNO2 reconstruction in space and the third order Runge-Kutta scheme with locally adaptive time steps. The model is validated against experimental data. Found analytically, that maximum run-up height of single waves of positive polarity on a beach of a conjoined water basin depends on the wave front steepness at the toe of the bottom slope. This dependence is general for single waves of different amplitudes and periods and can be approximated by the power fit: R max / R 0 = ( s / s 0 )^0.42. This dependence is slightly weaker than the corresponding dependence for a sine wave, proportional to the square root of the wave front steepness. The stronger dependence of a sine wave run-up on the wave front steepness is consistent with the philosophy of N-waves. Shown numerically, that all numerical curves for different wave amplitudes and periods, are parallel to the analytical one. Therefore, for estimates one can use the analytical dependence on the wave front steepness.