Answer: You can certainly cut back the dead parts of the branches until you find live wood. Ordinarily we recommend not pruning off more than one third of the live growth in any one pruning session to help avoid stress to the tree. So, start by cutting back the branches in increments until you find live wood. Do this all over the tree. Once you've reached live wood you can then prune the tree for shape, removing less than one third of the live wood overall. The sprouting from the trunk indicates the tree knows most of the top is dead and it is trying to survive by sending up new stems and leaves. I hope your tree recovers!
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