About a year back I posted a field guide of recursion schemes on this blog and then lost it a few months later when I lost a couple of months of blog entries to a crash. I recently recovered the table of recursion schemes from the original post thanks to Google Reader's long memory and the help of Jeff Cutsinger.

The following recursion schemes can be found in category-extras, along with variations on the underlying themes, so this should work as a punch-list.

Folds
Scheme Code Description
catamorphism† Cata tears down a structure level by level
paramorphism*† Para tears down a structure with primitive recursion
zygomorphism*† Zygo tears down a structure with the aid of a helper function
histomorphism† Histo tears down a structure with the aid of the previous answers it has given.
prepromorphism*† Prepro tears down a structure after repeatedly applying a natural transformation
Unfolds
Scheme Code Description
anamorphism† Ana builds up a structure level by level
apomorphism*† Apo builds up a structure opting to return a single level or an entire branch at each point
futumorphism† Futu builds up a structure multiple levels at a time
postpromorphism*† Postpro builds up a structure and repeatedly transforms it with a natural transformation
Refolds
Scheme Code Description
hylomorphism† Hylo builds up and tears down a virtual structure
chronomorphism† Chrono builds up a virtual structure with a futumorphism and tears it down
with a histomorphism
synchromorphism Synchro a high level transformation between data structures using a third data structure to queue intermediate results
exomorphism Exo a high level transformation between data structures from a trialgebra to a bialgebraga
metamorphism Erwig a hylomorphism expressed in terms of bialgebras
metamorphism Gibbons A fold followed by an unfold; change of representation
dynamorphism† Dyna builds up a virtual structure with an anamorphism and tears it down with a histomorphism
Elgot algebra Elgot builds up a structure and tears it down but may shortcircuit the process during construction
Elgot coalgebra Elgot builds up a structure and tears it down but may shortcircuit the process during deconstruction

* This gives rise to a family of related recursion schemes, modeled in category-extras with distributive law combinators
† The scheme can be generalized to accept one or more F-distributive (co)monads.