I have a neighbor with same situation. She has resodded 4 times. All failures.

No sun - no grass. Won't matter the grass type or how many times you try.

Boulders, river rock, liropie. Done and done.

I'd say St. Augustine but you'll still need to trim the trees . If you're not really decided on grass ; there's a lot of shade-tolerant plants (ferns, hostas, azaleas, etc.) to choose from .

Just make sure you use someone that knows trees. You wouldnt believe how many sick tree calls i get because they had their "landscapers" trim the trees. You dont want to harm your trees in the process. They take alot longer to grow back than the grass does.

Here is some good info: Tree Pruning

Dwarf Tall Fescue. You can seed it now. Till or aerify, seed at 10 lbs/ 1000, drag it in, fertilize with 13-13-13, keep seedbed moist throughout day.
Thank me later.

PS Reseed every spring and fall at 5 lbs/1000 until you have a stand strong enough to withstand our summers. Mow at 2.5-3 inches.
Very easy and cheap option.
Keep trees pruned to improve stand of grass.

Following. The new house has almost zero grass in the back yard and with all the rain it was basically a mud pit for a couple days. Great big old pecan tree that's absolutely covered in nuts (and squirrels) and a huge old live oak means there's very little sun back there.

But we're just renting right now so I don't want to spend the coin to fix it right.

Look into St.Augustine Palmetto grass. It tolerates shade. You need to do the right prep.Take a soil sample to your Ag agent or office. See how healthy your soil is. Wait till spring. Usally after Easter. Spray yard with round up. You want to kill everything in your soil. Till the soil about 4 inches deep. Put your sod out. The 2 biggest problems with St.Augustine are grub worms, and brown patch. Brown patch is caused from to much water. It can be treated easily. Like I said prep work is very important. If you get someone to do it. Take your time and check on them.

Zoysia. Thank me later. It will tolerate the shade and needs much less mowing than other grasses.

Like others have mentioned, trim (thin out) your trees. St. Augustine Palmetto like someone mentioned will work with filtered sunlight. I have a huge live oak that covers my front yard. I keep it thinned out and the St. Augustine gets filtered sunlight in the morning through midday.

Like was also mentioned, get your soil tested. When I put in a new yard of St. Augustine,I normally till the area, rake out all old grass-weeds, put down a 2" layer of 50/50 or similar compost mixture. Each time the grass has taken off.

Depending on your nighttime temps, it might be a little late to go with sod. I will work but you will have very little root growth when the nighttime temps fall below 60-65. Rye will get you through until spring. Overseed the area and in about two weeks you will see new growth.

I took a closer look and we should be able get prune a few branches and get some decent sunlight there especially in the afternoon/evening hours. Shouldn't be as much as I was originally thinking.

I wouldn't mind getting rid of that cottonwood either. It's not helping anything and it could cause a lot more trouble than it's worth between all the leaves that are falling/cotton in the spring, and not to mention any ice storms to worry about. It's huge though so it wouldn't be cheap. Probably a good 4' at the base

I would get rid of the tree.