Just as "grass doesn't grow on a playground" it often has a real tough time growing under trees!
A combination of factors makes growing grass under trees extremely difficult. Lawn grasses not only have to compete with tree roots for moisture and nutrients, they also have a tough time getting enough sunlight. Therefore, there are several steps you can take to encourage lawn growth in the dry shade underneath trees.
Thin-out tree branches to allow the maximum amount of sunlight to reach the ground. You will need to repeat the thinning-out procedure every few years. Avoid "topping" trees if at all possible.
Mow the lawn at the tallest height possible without the grass laying-over. Longer grass blades are able to intercept more sunlight. Also, longer shoots mean deeper roots!
Water lawn areas beneath trees thoroughly once a week during dry spells. Tree canopies can shed rainwater away, making the soil beneath trees extremely dry. In addition to that, tree roots remove moisture from the soil, creating severe competition with lawn grasses.
In spite of proper thinning, mowing, and watering, shady areas under trees often need to be reseeded every year or two. It's often necessary to add just enough soil to create a seed bed, since the existing soil can be filled with fibrous roots. While it can be very detrimental to add soil over most tree's root zones, limited soil fill over some tree roots shouldn't cause any major problems.
The depth and extent of the soil fill should be limited as much as possible. Some tree species can be killed by just a couple of inches of soil over their root zones. Consult with a professional landscaper or arborist before adding soil to a tree's root zone!
In the example below, we were contracted to seed an area beneath a pin oak that had recently been thinned-out by a tree service. Due to the mature size and species of the tree, it was determined that adding a thin layer of soil to the root zone wouldn't cause any serious problems.
In most cases, tree roots will eventually find their way back to the lawn surface, and this sort of "fix" may only be temporary. If you have an extremely difficult area that seems impossible to grow grass, consider mulching the area or planting a shade-loving ground cover. Remember to check with a professional before adding soil to a tree's root zone.