There is a very simple way to mulch around trees without killing the grass with glyphosate, digging up sod or using landscape cloth or plastic to smother the grass. Use newspaper instead! Four to six pages of newspaper with mulch on top will effectively kill the grass. Your trees will love not having to compete with grass for nutrients and water and the newspaper will decompose and continue to feed the tree. Newspapers use soy-based inks so there is no fear in harming the earth!
See how easy:
And voila! Ten minutes later: