you might want to thatch your yard first, get all the old dead grass out, and then you’re going to want a dusting of new topsoil not much, just like an 1/8 inch, throw down your new seed, starter fertilizer, and probably some type of lime, because if your dog is back there pissing and shitting, the ground is probably somewhat acidic, a lime will help to neutralize that, then on top of all that throw down a tiny bit of mushroom manure, once again, like an 1/8 inch, just enough to cover the seed to keep the birds out.
Thats the best way to take care of the problem, and trimming the trees would help as well. I’m a foreman for a landscaping company, we do it all the time.