Question: Why do I have trouble keeping my St. Augustine grass growing under my trees? Also, what is this clover-like plant that is killing my grass?

Answer: The attached images show that your St. Augustine grass has a virus called St. Augustine Decline (SAD)--you can see the mottling on the leaf blades. This is why Floratam St. Augustine should be planted or sprigged into the area--it is SAD resistant, resistant to chinch bugs and is drought tolerant. See: http://www.plantanswers.com/grass.htm

The "clover" you have is oxalis which is a shade-tolerant, weed/groundcover which invades as your turfgrass thins. You can spray it with a herbicide which is safe for St. Augustine such as Greenlight Wipe-Out but it is difficult to control and the application may further weaken the St. Augustine. No grass grows well in a shaded area---St. Augustine will endure it longer than most.

Another "weak grass invader" image is attached. It is the wide-leafed, tiny yellow bloomed ground cover named Horse Herb or Sprawling Horse Weed (Calyptocarpus vialis Less.) which is even harder to kill than oxalis. It too takes advantage of thinning turf.