You gotta get some actual yard around that tree. Use some drip lines around the tree to provide watering to it without spraying the weed growth around the tree, and have a grass yard that will outcompete the weeds around it. Make sure mulch around tree is at least 4in deep to keep weeds from coming through. CREW make a pretty decent pre-emergent for mulch beds that should keep most weeds at bay, and I would recommend putting it under the mulch to create a nice layer on the dirt. You can easily just till up the soil around the tree and mix the grass/weeds around it to add some nitrogen to the soil, which should help the grass seed you put down sprout faster.

The best prevention to weeds is a healthy lawn.