We study thermoelectric power under strong magnetic field (TPM) in carbon nanotubes (CNTs) and quantum wires (QWs) of nonlinear optical, optoelectronic, and related materials. The corresponding results for QWs of III-V, ternary, and quaternary compounds form a special case of our generalized analysis. The TPM has also been investigated in QWs of II-VI, IV-VI, stressed materials, n-GaP, p-PtSb2, n-GaSb, and bismuth on the basis of the appropriate carrier dispersion laws in the respective cases. It has been found, taking QWs of n-CdGeAs2, n-Cd3As2, n-InAs, n-InSb, n-GaAs, n-Hg1?xCdxTe, n-In1?xGaxAsyP1?y lattice-matched to InP, p-CdS, n-PbTe, n-PbSnTe, n-Pb1?xSnxSe, stressed n-InSb, n-GaP, p-PtSb2, n-GaSb, and bismuth as examples, that the respective TPM in the QWs of the aforementioned materials exhibits increasing quantum steps with the decreasing electron statistics with different numerical values, and the nature of the variations are totally band-structure-dependent. In CNTs, the TPM exhibits periodic oscillations with decreasing amplitudes for increasing electron statistics, and its nature is radically different as compared with the corresponding TPM of QWs since they depend exclusively on the respective band structures emphasizing the different signatures of the two entirely different one-dimensional nanostructured systems in various cases. The well-known expression of the TPM for wide gap materials has been obtained as a special case under certain limiting conditions, and this compatibility is an indirect test for our generalized formalism. In addition, we have suggested the experimental methods of determining the Einstein relation for the diffusivity-mobility ratio and the carrier contribution to the elastic constants for materials having arbitrary dispersion laws.