A lot of people think our biggest fear when backpacking should be bears or snakes or other mainstream phobias; but personally, falling trees are always my biggest concern. I see enough fallen and weakened trees when hiking and backpacking to realize that I have little control over such a catastrophe.
Some thoughts to avoid such a situation:
-Overused sites. Lots of hatchets, debarking, and campfire damage are done to trees in such sites that make them potentially weaker to storms/winds.
-Creaking noises. A few weeks ago, I came across a creaking noise while trail running along a trail I frequent. I concluded the noise to be coming from an apparently healthy tree and continued on. I came back to that location 2 days later and found that the tree had fallen over.
-Sites below tree-covered slopes. You'll find that most fallen trees are along sloped locations. The root-structure of these trees obviously are not as solid, and usually they will fall downslope. Avoid camping below such locations.
All seem pretty obvious, but at the end of a long/tiring day we tend to overlook such details when setting up camp. I'm not as concerned about such things when active because I feel the environment will give me fair warning, but, as always, prudence should be taken when hiking in severe conditions.