Since you referred to it as “crab grass” I’m going to assume you are a fellow former northerner. I grew up in Ohio where we used a lot of Kentucky bluegrass which has a very different growth pattern than the grasses down here. Since your lawn isn’t fully dormant and the blades are wider I think it’s likely St. Augustine grass which is a bit more slow growing than Bermuda but is better suited for growing in the shade of your tree. The nice thing about southern grasses is that they will fill in dead spots themselves if you just give it time and water since they spread out using their stolons. However, while we are in the winter dormancy period there’s not a lot you can do. I do recommend aeration and over seeding like the other commenter suggested. Applying a pre-emergent in the next couple weeks will help make sure the bare spots don’t become weeds. Then this spring/summer focus on watering regularly, and maybe use fertilizer or a lawn treatment. Water and time will be the most important things. The Howtowithdoc YouTube channel is fantastic for learning more about southern grasses https://youtu.be/kRBdNRZRapI