Our property has a lawn which is bordering a wooded area. Somewhere
in the wooded area is a tree which has extended its roots under our
lawn (and our neighbours' lawns). This root system manifests itself
by poking tiny "branches" up the lawn with leaves that are plainly
visible. So when I see this I dig down with the shovel and pull up
as much of the root as I can. This of course, rips up the lawn which
makes it look ugly.

So what can be done about this?

Are you certain this is a tree and not something else? From your
description it sounds very much like bamboo. Got a picture of the roots
and/or shoots?

James