Trollshaws are the upland woods, consisting at least partly of beech trees, that lay to the west of Rivendell between the Rivers Hoarwell and Loudwater. They were the haunt of Trolls, three of which famously waylaid Bilbo and his companions during the Quest of Erebor. Years later, Frodo and his companions found them on the way to Rivendell, but the Trolls, from the distractions of Gandalf, had been turned to stone by the sun. On the hills of the Trollshaws were shallow caves, and on the top of some of the hills there were castles and towers that had been built by the Men of Rhudaur.
It seems there were few if any Trolls that lived in the shaws by the late Third Age. They were probably driven back into the Ettenmoors (which was Troll country) to the North, probably by the Dúnedain Rangers and the Elves of Rivendell.
Shaws is an archaic word meaning "thicket, wood"