This very green part of our grass lies directly underneath our crabapple tree. Every fall, it is covered with decomposing crabapples. The rest of the yard does not get this special treatment.
As you can see, that organic material really does the job.
Our plan is to reduce the lawn in the back by creating vegetable & flower beds all around the perimeter. So far, though, we've focused most of our efforts on the front yard. (Click that link to see a photo how-to of sheet composting.) Here's what the flower patch looks like today.
We put wildflower seeds down so our neighbors didn't have to look at a yard full of hay all summer (hay is the top layer of the sheet composting). Also, some of the seeds we planted are cover crops with good strong root systems that should help to loosen up the soil.
And the pumpkin plants that have come up out of the compost are growing well. I'm sure lots of people wouldn't find this to be a good thing. But I'm thrilled.
As you can see, I enjoy a bit of chaos in the garden.