I have read online and heard from a number of people that large shade trees take up quite a lot of water, which can help with situations where part of a lawn is excessively wet. 

Is there any truth using trees as a solution for wet areas of the lawn, or if an area is excessively wet it will likely continue to be excessively wet after a large shade tree grows in the area?

Also, while I have no doubt that large trees can take up quite a lot of water, wouldn't grass (if it's not so wet to grow it there) have equal or more water uptake and evaporation rates than a tree?