Granitecitygirl, I appreciate the comment about the fact that MOST labradoodles shed, as they do, however I do need to correct you. The non-shedding coat is not recessive . . . nor is it dominant. It depends on the circumstances of other genes.
Much like coat color in dogs is not always simple, there are multiple genes involved that have to come into play to allow non-shedding and shedding. The "bearded" gene, which allows for the continuous growth of a coat, only works consistenly when matched with double recessive "long coat" alleles (as opposed to the dominant short coat allele) and is actually dominant in that circumstance. When you breed a short coat dog that sheds - lab - to a long coat dog with a bearded gene - poodle - things get very difficult to predict and there is a variety.
Now if you breed a single coated "shedding" dog with two "long coat" alleles to a dog with a bearded gene and a single coat, then you can count on all the offspring having a continuous growth coat (non-shedding, hypoallergenic is a whole different ballgame though). An example I can think of is the Cavalier crossed with the Poodle. Both breeds have already set long coat alleles and single coats, and the bearded gene then works as if it is dominant. As most breeds have double coats, it is rarely this easy though.
By the way I am merely a lifetime dog lover who studies human genetics, so naturally this was a branch out. I have a friend who has done a thesis in canine coat color genetics who explained the above information to me, and as well explained that not all the genes at play have been studied well enough yet to understand them completely.