Phylogenetic taxonomy of the Vertebrates (mun.ca) has the following graphic (14 clades total, minimum of 12 collapsing Amphibia):
 becomes 6 clades, and 'Reptiles' (the paraphyletic group of non-avian Sauropsids) becomes 3.
It seems the question is equivalent to asking "what is the minimum number of clades necessary to partition Vertebrates while maintaining 'birds' as a clade?".