Recent experimental measurements on transport through carbon nanotubes in the Colomb Blockade regime showedm at zero bias and zero magnetic field, an alternation in the Coulomb peak heights as electrons are added to the nanotube. This surprsing peak-height alternation is shown to be related to the odd/even parity of the number of electrons in sider the nanotube. We invesitiaged this phenmenon theoretically with a new many-body apporach. and find that the anomalous behaviour may be attributed to the spin degenerate states. We alos show that asymmetric current saturation as a function of bias voltage polarity, also observed experimentally, is due to eigenstate relaxtions within the nanotube.