A theoretical study of the quasistationary process of stable growth of an anisotropic dendrite under conditions of convective heat transfer in a melt is carried out. Based on the theory of morphological stability and the theory of microscopic solvability, a selective relationship is derived for the stable growth rate of the dendrite and the diameter of its vertex. The analysis of the selected ratio is performed. Quasistationary solutions describing the temperature distribution in the liquid before the growing parabolic plate (the two-dimensional shape of the needle-like dendrite) and the paraboloid of rotation (the three-dimensional shape of the needlelike dendrite) are determined.