Trees for paper are grown specifically for that...
...so they aren't cutting down forests to make paper products. However, I don't know if making a paper plate is more wasteful than washing and disposing of a plate. During the life of a plate, you use an awful lot of water, detergent and energy to heat the water that you use for washing that plate. The dish-washing liquid is in plastic bottles, which are petroleum products, and the plate doesn't biodegrade when it's eventually discarded, so I'm tending to think that paper plates may actually be better for the environment, but I don't know for sure.