Walk around the partially fallen tree to evaluate its most stable points. If you cut a tree that is not stable or in a sturdy position, the tree to snap toward you or fall at awkward times, possibly injuring you.
Limb the tree, which means cut off all the limbs before moving on to the trunk. Stand on the opposite side of the trunk from the limb to be removed. If possible, stand uphill from the limb and cut in sections. Plan an escape route for yourself in case the limb rolls upon being released from the tree when you cannot stand on the opposite side of the trunk and safely cut the limb.
Place yourself uphill from the tree after the limbs are removed. Cut the trunk of the tree in sections, starting with the section closest to the ground. Plan an escape route for yourself in case the trunk falls in your direction. Cut a section at a time until the entire tree is chopped into sections.
Divide the sections into easy to move pieces. The pieces need to be small enough to lift safely; typically this is less than fifty pounds for a healthy adult. Again, make sure to cut larger pieces while standing uphill to avoid being injured when they roll.