When I say (c,n+1) has a "living" father what I mean is a father (b,n) such that b is alive in generation n.
Your father might have died before you were born but he didn't die before he himself was born.
I handwaved that bit anyway for expediency, if you're interested in the full details see: http://www.combinatorics.org/ojs/index.php/eljc/article/view...
Okay, I read that and now I get it. Each live cell has at least three ancestors. Two or fewer of these ancestors come from a forbidden direction, so at least one comes from an allowed direction. Among all of the ancestors from an allowed direction, arbitrarily choose one to be the “mother”. All of the other ancestors are called “fathers”. There's no requirement that a father has to come from a forbidden direction.
This statement seems incorrect. It's true that every living cell has a living mother, but there's no requirement of a living father.