{-----------------------------------------------------------------
Overview: A "Tree" represents a binary tree. A "Node" in a binary
tree always has two children. A tree can also be "Empty". Below
I have defined "Tree" and a number of useful functions.
This example also includes some challenge problems :)
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data Tree a = Node a (Tree a) (Tree a) | Empty
empty = Empty
singleton v = Node v Empty Empty
insert x tree =
case tree of
Empty -> singleton x
Node y left right ->
if x == y then tree else
if x < y then Node y (insert x left) right
else Node y left (insert x right)
fromList xs = foldl insert empty xs
depth tree =
case tree of
Node v left right -> 1 + max (depth left) (depth right)
Empty -> 0
map f tree =
case tree of
Node v left right -> Node (f v) (map f left) (map f right)
Empty -> Empty
t1 = fromList [1,2,3]
t2 = fromList [2,1,3]
main = flow down [ display "depth" depth t1
, display "depth" depth t2
, display "map ((+)1)" (map ((+)1)) t2
]
display name f v =
text . monospace . toText $
concat [ show (f v), " ⇐ ", name, " ", show v ]
{-----------------------------------------------------------------
Exercises:
(1) Sum all of the elements of a tree.
sum :: Tree Number -> Number
(2) Flatten a tree into a list.
flatten :: Tree a -> [a]
(3) Check to see if an element is in a given tree.
isElement :: a -> Tree a -> Bool
(4) Write a general fold function that acts on trees. The fold
function does not need to guarantee a particular order of
traversal.
fold :: (a -> b -> b) -> b -> Tree a -> b
(5) Use "fold" to do exercises 1-3 in one line each. The best
readable versions I have come up have the following length
in characters including spaces and function name:
sum: 16
flatten: 21
isElement: 46
See if you can match or beat me! Don't forget about currying
and partial application!
(6) Can "fold" be used to implement "map" or "depth"?
(7) Try experimenting with different ways to traverse a
tree: pre-order, in-order, post-order, depth-first, etc.
More info at: http://en.wikipedia.org/wiki/Tree_traversal
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