Ask Mr. Smarty Plants

You did not say what grass you or your  neighbors have. The rings might be a problem in the grass rather than originating from the tree. We will investigate brown rings in non-native grasses bermudagrass, St. Augustine and zoysia.

Although no grasses do well under trees, oak trees have more of an influence on the grasses. Oak, walnut, pecan and other trees exhibit allelopathy which means they exude substances harmful to competitive plants beneath them. Here is a a previous Mr. Smarty Plants answer on how we feel about lawns under trees. Between the shade, the allelopathy and the competition of extensive root systems for water and nutrients, grass growing under trees is always difficult. Since all three of the grasses we mentioned above are non-native to North America we don't know much about them, but will try to find some research to help you make a diagnosis.

Bermuda Grass: From WalterReeves.com Bermudagrass diseases - brown patch.

Zoysiagrass: From Missouri University - Issues of Zoysiagrass Lawns

St. Augustine Grass: From Cypress Isle (FL) Master Gardeners- St. Augustine Grass, Common Summer Problems.

Of course, if you are growing Bouteloua dactyloides (Buffalograss) or Habiturf they both need at least 5 hours a days of sun and won't be growing under the trees anyway.