We introduce an adaptation of the well-known tree+SPH numerical scheme to post-Newtonian (PN) hydrodynamics and gravity. Our code solves the (0 + 1 + 2.5)PN equations. These equations include Newtonian hydrodynamics and gravity (0PN), the first-order relativistic corrections to those (1PN), and the lowest order gravitational radiation terms (2.5PN). We test various aspects of our code using analytically solvable test problems. We then proceed to study the 1PN effects on binary neutron star coalescence by comparing calculations with and without the 1PN terms. We find that the effect of the 1PN terms is rather small. The largest effect arises with a stiff equation of state for which the maximum rest mass density increases. This could induce black hole formation. The gravitational wave luminosity is also affected.