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At school I was taught the Linnaean view of vertebrate life - that vertebrate animals consisted of 5 distinct groups: mammals, birds, reptiles, amphibians, fish.

I learned later however that this was phylogenetically inaccurate, with birds being 'closer related' to crocodiles than other reptiles are, for example.

As such I am curious - since the idea of these groupings is so strong in popular conceptions of animals:

What is the minimum number of clades we would need in order to partition 'Vertebrates', maintaining 'Mammals', 'Amphibians', and 'Birds' as clades?

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    $\begingroup$ @Remi.b By completely partition I mean, every member of vertebrates belongs to one of the clades, and the clades are disjoint; by minimum I mean make the remaining clades as large as possible given the constraint Birds, Mammals, Amphibians are already clades. e.g. Crocodilians is 1 clade (considering smaller clades e.g. Crocodiles, Alligators & Caimans just increases the amount we need to consider, therefore not minimum), and it cannot be extended to form a larger clade as this would be (at least) 'Archosaurs', and hence absorb 'Birds', breaking one of the conditions. $\endgroup$ – brazofuerte May 23 '18 at 22:28
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    $\begingroup$ Oh I got it now $\ddot \smile$. Here is the beginning of the tree of vertebrates and as you can see there are already quite of few branches that will need named to "completely partition the clade of vertebrates". One could just screen through tolweb.org (or some other similar ressource) and count. $\endgroup$ – Remi.b May 23 '18 at 22:39
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Phylogenetic taxonomy of the Vertebrates (mun.ca) has the following graphic (14 clades total, minimum of 12 collapsing Amphibia):

enter image description here

Here, what was preivously 'Fish' (the paraphyletic group of non-tetrapod vertebrates) 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?".

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