Why did China feel it was necessary to invade Tibet?

700 years ago, Tibet was under Mongolian's control without much force. China was ruled by the same emperor during that period. 300 years ago, both Tibet and China were under Manchu's control. 100 years ago, Chinese overthrew Manchu emperor and established republic. As Tibet had been in the same empire for centuries, Chinese thought that Tibet was a part of China. So when the Communists took the power in China 60 years ago, they sent troop to Tibet and took that over. You asked why, and the reason is that they believe that Tibet is a part of China. Kind of like why UK invaded Falkland Islands -- UK believes that that belongs to UK.