Dan Gill has some excellent advice on how to handle damaged trees from storms.
Trees are particularly vulnerable to the effects of high winds and sustain various types of damage as a result. Much of the work dealing with trees after a hurricane should be done by professionals who have the equipment and training to do the job safely. Most people do not have the equipment or expertise to safely remove large trees or fallen trunks. Also remember that downed power lines are often present around fallen trees, so take necessary precautions.
After a storm, you should have fallen trees removed as soon as it is practical (obviously, trees that fall on or near the house get top priority). If you want the stumps ground out, you need to check with the company removing the trees. The ground-up wood left from the stump grinding (and also ground-up branches) may be used as mulch in planting beds, for covering walkways or composted for use as a soil amendment during bed preparation.
Homeowners can handle smaller trees and branches with chainsaws. This may be necessary to clear pathways or driveways or remove branches around a home. It is critical that you understand the safe use of such equipment and follow the manufacturer's safety precautions carefully. Read user instructions thoroughly and do not attempt to tackle jobs that are beyond your ability to carry out safely.
If you have damaged trees, remove large branches that are broken but still hanging. These branches pose a significant risk because they can fall at any time. This should be taken care of as soon as possible. Less critical are broken stubs where branches were lost. These stubs, however, should eventually be pruned off to allow the trees to heal over the wounds. Make the final pruning cut just outside of the branch collar at the base of the branch.