The aim of the present work is to analyze free flexural vibration and buckling of single-walled carbon nanotubes (SWCNT) under compressive axial loading based on different constitutive equations and beam theories. The models contain a material length scale parameter that can capture the size effect, unlike the classical Euler-Bernoulli or Reddy beam theory. The equations of motion of the Reddy and the Huu-Tai beam theories are reformulated using different gradient elasticity theories, including stress, strain and combined strain/inertia. The equations of motion are derived from Hamilton's principle in terms of the generalized displacements. Analytical solutions of free vibration and buckling are presented to bring out the effect of the nonlocal behavior on natural frequencies and buckling loads. The presented theoretical analysis is illustrated by a numerical example, and the results are qualitatively compared by another results.