Why did President Truman order an invasion of North Korea?

He did not. In fact this may be the only time the UN, did anything of importance. It was a UN declaration to go in and stop the communists that ultimately led to a multi-national force or Aussies, New Zealanders, Canadians, British, Americans, French, and a few other countries I cannot remember. Now President Truman took a case before the UN, but the UN made the multi-national invasion order.