Turf Lack of Sunlight

I have a large Linden tree and the turf under it does not grow well.  I think it because the tree is taking all the water.  What kind of fertilizer do I need to make it grow?

The thinning turf is probably not from lack of water, but lack of light. You could try a shade blend turfgrass mixture, which might include shade tolerant cultivars of Kentucky bluegrass, turf-type tall fescue, or red fescue.  You can also limb-up the tree a little to allow more light into the turf.  To grow well, turfgrass needs a minimum of 6 hours of direct sunlight per day.