The best-case scenario would be only small, broken branches need removing. You can perform this yourself, but take care not to leave an opening in the tree that will allow pests to enter the tree. Following storm damage your tree needs extra strength to heal, an invasion will only add to its problem.
In some circumstances its possible to wrap a branch, allowing the wood to fuse. If the branch is held on by bark alone, it’s already too late. Likewise, if the wound has dried out, it may be best to cut off the branch so a new one can fill its place.
For large branches, we always recommend using a specialist. A professional tree surgeon will have a range of options at their disposal and furthermore, have the necessary insurance and expertise to carry out the work safely.