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Monophyletic: a group of taxa comprised of a shared common ancestor and
all of its descendants.
Paraphyletic: a group of taxa comprised of a shared common ancestor but
NOT all its descendants.
Polyphyletic: a group of taxa which does not also include their common
ancestor.
For
example, take the tree: mammals + (turtles + (lepidosaurs + (crocodilians +
birds)))
"Reptlia" as traditionally concieved is turtles, lepidosaurs, and
crocodilians. Their shared common ancestor is also considered a
reptile. However, this grouping excludes birds (because they are so
radically transformed). Thus, the traditional "Reptilia" is
paraphyletic.
Reptilia as used by Gauthier (and many others now!) comprises turtles,
lepidosaurs, crocodilians, birds, their common ancestor, and all of its
descendants. This grouping is a monophyletic grouping.
Some
workers have proposed a taxon "Haematothermia" (the warm-bloods) for birds plus
mammals, but nothing else. Given the tree above, the common ancestor of
the haematotherms would NOT also be a haematotherm, and so Haematothermia is
polyphyletic.
Hope
this helps.
Thomas R. Holtz,
Jr.
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