That could also be caused by a lack of water for the grass. Big trees tend to **** up a lot of moisture in the ground surrounding them. This takes water from the grass causing it to turn brown and thin out. I work at a golf course where this was VERY noticable this July and August, because of a lack of rain and no irrigation in the rough.I don't have, nor want to buy a spreader....The problem is, I have a giant oak tree providing shade. The grass is nice outside the shadow, inside the shadow, it is not as green/thick
If its possible, you could try watering that area every day for a half hour or so. I'd hook up a hose and a sprinkler. Hand watering = SOOOOO BORING, even with music.
