My son kept deleting apps he decided were "yucky" haha...
Here is how you stop them from deleting apps you want.
Go to Settings -> General -> Restrictions
You will now be prompted to enter a pin so that your child can't come back and turn the restrictions off.
Now find deleting apps and swipe it to the left to turn it off.
Now if you want to delete an app, you have to come back here and turn it on. It's a pain but its better then trying to figure out what app they deleted!