Some things that can be done so you can get grass to grow under trees is to trim them a little to allow more sunlight to get through. If we get a DRY summer you may need to water these areas to protect them from drying out since the trees when they need water suck everything they can around it to survive. In the fall its always good to toss some seed in these shady areas to get new growth always trying to come in and you stay ahead of it.