We apply the cummulant method to obtain the high-temperature expansion of the Helmholtz free energy of the tetrahedral spin-1/2 and spin-2 XXZ models. The tetrahedral model is written as a composite spin-1 XXZ model, and some of its thermodynamic functions are compared to those of the ordinary spin-1 XXZ model. The composite spin-1 model is then mapped onto a fermion model, and it is shown that the contribution of the string Hamiltonian to thermodynamic functions at high temperatures cannot be neglected. The high-temperature expansion of the Helmholtz free energy of the anisotropic spin-
2 XXZ chain is obtained up to order ${\ensuremath{\beta}}^{6}.$ Our results fit well the numerical quantum Monte Carlo data calculated by Yamamoto [Phys. Rev. B 53, 3364 (1996)] for the isotropic antiferromagnetic Heisenberg chain. We complement his high-temperature expansions for thermodynamic functions with terms of higher order in $\ensuremath{\beta}.$