If you find dead wood, it must be removed immediately, especially if it's an overhanging branch. If it is over a sidewalk you might want to block the walk until you get the problem resolved.
Inspect the tree for cracks or split in the trunk. High winds might cause a split to occur where two trunks of the tree meet or where a branch grows from a trunk. This split or crack means the structural strength of the tree is weaker.
This type of damage should be inspected by a tree specialist. A trained arborist can evaluate a tree to determine whether it can be salvaged or should be removed.
Also, look at your tree for signs of decay. If you have a cavity in the trunk of the tree, is the wood soft and decayed inside? If it is, again, get a trained professional to see if the tree is still structurally sound.
Look for signs of rot at the base of the tree. If you notice fungi growing on the trunk or around the base of the tree, it probably means that there is decay happening to the trunk or the roots and further evaluation by an arborist is important.
While you are looking at the base of the tree, are there girdling roots visible or damage from string trimmers? These damages can result in a weaker tree and can often be part of the reason why a tree fails. I see trees in the area that people try to keep growing that are structurally so bad that they should be removed. These are trees where the weight of the tree is uneven and off to one side of the trunk or a tree that leans excessively.
If you have any doubts about a tree in your yard, consult a professional. If the tree needs to be removed, they can do this for you. Perhaps the weakness can be corrected with cables and bracing, or by proper pruning. Never let someone come in and remove the top from your tree and end up with large stubs where branches used to be. This is extremely hard on the tree and you will never improve your tree with such a practice.