I think Obama can be tried for treason because he was handed a passport when he was a US Senator. This was his first American passport. It should legally not have been given to him, but it was, for diplomatic reasons (he needed to travel out of the country on official senate business).
So becusse of that gift of a US Passport, he is now an American citizen.
He can be tried for treason because he took the oath of office as a senator and twice as president. We have foreign nationals in our armed services who all take the oath to protect and defend the Constitution when sworn in to service. They can certainly be tried for treason precisely because they took that oath.