It sounds like not enough water. If it is thin under the tree and no grass at all out in the sun, then the cool shade of the tree is what is saving that little bit of grass you have. Are you willing to water all summer? And when I say, water, I mean deep watering, one inch, all at one time, as often as once per week? If you are willing to water, then you can have a nice lawn all season.

Here is what I would do with that. I would test the waters to see if you want the effort of a nice lawn. First off you should know that spring is a poor time to seed a new lawn. Late summer to early fall is the very best time of year for that. Second you should know that all the normal turf grasses need about the same amount of water every week. Third, there are some grasses which do well in the shade and some that will not grow at all in the shade. Okay, in order to test this lawn project, I would start to water the entire lawn with an oscillating type sprinkler. I'll have more on watering at the end of this. Oscillators provide the most even coverage, so, to me, they are the least hassle to figure out and result in the easiest for a beginner. Once you start watering you will get both grass and weeds growing. I would not spray the weeds at all this year until late summer when you would be seeding. Why? Because the most important thing you can do is get roots growing to loosen the soil. Weeds will do that job perfectly. All you have to do then is mow them down once a week. Assuming the lawn has been neglected for years, I would apply an organic fertilizer now. Milorganite is the easiest one to find, but many of us use grains from the feed store in addition to, or instead of, Milorganite. Your choice. Then spend the rest of the summer enjoying infrequent watering and mowing. Don't do anything else to it. This is practice for the real thing next year. Then in August, write back here to get details as to reseeding your lawn. You will not need more topsoil. The roots in the ground and the organic fertilizer will really help your soil. And don't spend money on an irrigation system until you are certain you want to continue this project and turn it into a hobby.

Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.

neighbor to left has weed program. lawn is very nice. neighbor to right does his own weed stuff. i think just the spring application. his lawn is better than mine