A potential energy function comprising a two-body potential term which is modified form Morse potential and a three-body potential term which is modified from Axilrod-Teller potential has been developed for small carbon clusters. The structural changes of small carbon clusters $C_2-C_6$ are qualitatively investigated by employing this potential energy function representing the energies of the small carbon cluster isotopes as a function of the three body intensity factor. It is found that the structure of the small carbon cluster changes from open structure to closed one, from complicated structure to simple one, and from three-dimensional structure to two-or-one dimensional one as the degree of the three-body interaction increases.