Any plant poison that would kill the grass will also do harm to the tree. However, if you put down something like newspaper or cardboard (to keep the grass from getting the sunlight it needs to grow) and then cover that with gravel that will control the grass fairly well with little harm to the environment.

kimmq is kimmsr

Some weed killers will only kill narrow-leaf plants such as grasses;

others kill only broad-leaf plants such as trees, and shrubs;

and still others are non-selective in that they will kill both narrow-leaf plants and broad-leaf plants.

"Any plant poison that would kill the grass will also do harm to the tree. "

This is incorrect. There are indeed selected herbicides that target monocots or grass-like plants only - Grass-Be-Gone is a common product that is widely available. It can be applied with no issue to any larger growing, non-grasslike plant.

I also object to the terminology of "plant poisons" as opposed to herbicides. The first connotes a distinctly negative implication which is not always appreciated or true. While I do not advocate their use willy-nilly, herbicides do play a significant positive role in the environment if they are selected and used with care and according to label directions.