Exact solutions to the diffusion equation are obtained which correspond to diffusion controlled growth of elliptical paraboloids (dendrites; forward growth with linear time dependence) and spheroids (growth with square root of time dependence). The elliptical dendrite includes as special cases the dendrite of circular cross-section and the dendrite platelet. The dependence of the velocity V of dendritic growth on (the smaller) tip radius of curvature R, undercooling ΔgJ and axis ratio 1/√(1 + B) of the elliptic cross-section is found to be of the form V ∝ (Δϑ)nR, where the exponent n, a slowly varying function of Δϑ, is (contrarily to past assumptions) noticeably greater than 1 in the range of experimentally accessible undercoolings. The value of n is about 1.2 for the circular cross-section (B = 0), and about 2 for the dendritic platelet (B = ∞). The spheroid solution includes as special cases the known results for the sphere, the circular cylinder and the plate.