graphs and trees again

[Ross Paterson]

On Tue, Jan 13, 2004 at 09:36:58PM +0100, Wolfgang Jeltsch wrote:
> Am Dienstag, 13. Januar 2004 19:12 schrieb Ross Paterson:
> > * made the types abstract, but provided the equivalent constructor and
> >   destructor functions.  I'm not sure that gains anything.
> 
> Well, I did this to be consistent with the Graph module where abstract types 
> are crucial.  Yes, it doesn't make much sense.  But declaring Forest via 
> newtype instead of type surely makes sense because this way you're able to 
> define a meaningful Functor instance.

You get to write fmap instead of map . fmap, but you lose the ability
to treat forests simply as lists.

> > * added upwards and downwards accumulators (except that your "upwards"
> >   accumulator is actually a variant downwards accumulator, with quadratic
> >   performance).
> 
> What do you mean with "a variant"? Is foldl a variant foldr?

A downwards accumulation normally replaces each value with the foldl
of the path (list) of items from the root to here (as yours does).
An upwards accumulation normally replaces the values each item with
the (tree) fold of that subtree:

	foldTree :: (a -> [b] -> b) -> Tree a -> b
	foldTree n (Node a ts) = n a (map (foldTree n) ts)

	upAccumTree :: (a -> [b] -> b) -> Tree a -> Tree b
	upAccumTree f = foldTree ft
	  where ft a ts = Node (f a (map root ts)) ts
		root (Node a _) = a

It's a different generalization of scanr: yours replaces each item with
a foldr on the list of ancestors.

BTW, do you have any uses for these things?  (The unfolds are useful
for search problems.)