If you have a computerized social identification and social security system in your country you can try and contact the relevant agencies in order to track down their father. They normally have all the information stored in their system including their current whereabouts, credit history, profession, passport number and a permanent address which they may have submitted at the time of registration. In case your father has deceased, they also have that information stored in their system and essentially keep a track of every citizen. If your father has left the country you can check the exit list to confirm that. However, you will need some supporting document to help you out in your pursuit like say a social security number, high school diploma, copy of the driver's license or any such official document. There are some independent investigators as well who provide private consultancy in such cases.
You know urself whether he is dead or alive.
There are many charities who can help you try to trace him. Start with the Salvation Army who can advise you further.
You can hold a mirror in front of his nose or mouth if it doesn't steam up it means he is dead.