Most of the little Pacific Islands, of which Easter Island was just one example were settled by Polynesians, who, over a few centuries, would fell all the trees to make canoes. However, complete deforestation would destroy the soils, and eventually they would be left with no boats, no soil, no food, and no escape.
Diego Garcio in the Indian was once inhabited by about 1300 people, until Great Britain and the US decided they wanted to build a naval base there. The British, who had purchased it as part of the British Indian Ocean Territory leased it to the United States in exchange for Polaris Missiles. The inhabitants meanwhile were forcibly relocated to Mauritius, and never reimbursed. Some inhabitants are still making human rights claims, and wish to be settled in Britain itself.
Today the island houses a large naval base, US AFB, one of three NAVSTAR tracking stations, and is regarded by the US and UK to be exempt from the African Nuclear Weapons Free Zone.