Under which empire did ancient china control territories that are not a part of present day china

The Han dynasty, which has given its name to the main ethnic group in China, "the Han", was notable for its military might. The empire expanded westward to reach the Tarim basin (in the current Uyghur Autonomous Region of Xinjiang). Chinese armies also invaded and annexed parts of northern Vietnam and Korea towards the end of the 2nd century BC, but Han control of peripheral regions was generally scarce To ensure peace with local powers outside of China, the Han Court developed a mutually beneficial "tax system." Non-Chinese states were allowed autonomy in exchange for the symbolic acceptance of Han domination. The tributary ties were confirmed and reinforced through marriage links between the upper classes and the periodic exchanges of gifts and goods.

i honestly have no idea