ROFL, I love how you worded it.

Yes, it is normal for you to take damage if you get hit by falling trees. It is also normal for them to damage other trees (and potentially take them down) if they fall into other trees.

The key is to not get hit by the falling trees. Also, make sure you have eaten before logging, so that you have enough health that, should you fail at not-getting-hit, you do not die.