Could be some leaf scorch..Leaves nearest the sun are being affected while the leaves slightly shaded on the inner part are fine.The hottest part of summer is a prime time for scorch. Leaves transpire and must replace the lost water. If not enough water----leaf scorch. Grass could also be affected by lack of water and or lack of sun (shaded by tree). Try a little more water.
Mulch also helps hold moisture and may help both problems, tree and grass. Of course mulch might not please your personal taste.
Disclaimer: This is just my best guess. Keep an eye on the tree for any lack of vigor which could indicate other problems. If everything else looks good more water in summer should help going forward. Good luck

Yeah, that was my first guess. I was also thinking a nutrient deficiency, as the bare grass spots are the only bare spots and get almost full sun. The tree could be using all of nutrient x and not leave any for the grass.