If your deciduous trees have suffered storm damage, you should remove stubs, broken and hanging branches to minimize hazards and for future tree health. This is sometimes termed crown cleaning. Pruning cuts should be made according to good practice including leaving the branch collar and not making flush cuts. If possible, avoid removing branches that are greater than 1/2 the diameter of the branch or trunk left, 1/3 is preferred. Not following this practice can result in poor wound closure and sucker growth.
On older, slow-growing and drought-stressed trees, crown thinning and crown reduction that removes a lot of live wood and the stored carbohydrates in it can speed decline and are not recommended. No more than 25 percent of crown should be removed, and with many already weakened trees, no more than 15 percent. Younger, vigorous trees may more easily accommodate the loss of 25 percent. Consider pruning broken and dead wood now and delaying optional structural pruning for future years if pruning will result in too much loss.