Functional Programming Lecture 7: Trees, Study notes of Computer Science

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CS 457/557 Functional Programming

Lecture 7

Trees

Trees

• (^) Trees are important data structures in computer science.
• Trees have interesting properties:
  • They usually are finite, but statically unbounded in size.
  • They often contain other non trivial types within.
  • They are often polymorphic.
  • They may have differing “branching factors”.
  • They may have different kinds of leaf and branching nodes.
• (^) Lots of interesting things can be modeled as trees
  • lists (linear branching)
  • shapes (see text)
  • programming language syntax trees
• In a lazy language it is possible to have infinite trees.

Match up the Trees

• IntegerTree
• Tree
• SimpleTree
• List
• ITree
• FancyTree ‘a’ ‘b’ ‘a’ ‘b’ ‘c’ 2 6 9 ‘a’ ‘b’ 1 2 ‘a’ ‘b’ ‘c’

Functions on Trees

• Transforming one kind of tree into another: mapTree :: (a->b) -> Tree a -> Tree b mapTree f (Leaf x) = Leaf (f x) mapTree f (Branch t1 t2) = Branch (mapTree f t1) (mapTree f t2)
• Collecting the items in a tree: fringe :: Tree a -> [a] fringe (Leaf x) = [x] fringe (Branch t1 t2) = fringe t1 ++ fringe t
• What kind of information is lost using fringe?

Binary Search Trees

• InternalTrees (values at internal nodes) in sorted order.
• Used for efficient implementation of sets, dictionaries, etc.
  • Logarithmic access, update in average case data ITree a = ILeaf | IBranch a (ITree a) (ITree a) elemTree:: Ord a => a -> ITree a -> Bool elemTree v ILeaf = False elemTree v (IBranch w l r) | v == w = True | v < w = elemTree v l | v > w = elemTree v r

Building Search Trees

insertTree::Ord a => a -> ITree a -> ITree a insertTree v ILeaf = IBranch v ILeaf ILeaf insertTree v (IBranch w l r) | v <= w = IBranch w (insertTree v l) r | v > w = IBranch w l (insertTree v r) listToTree xs = foldr insertTree ILeaf xs s = listToTree [1,4,3,5,2,9,8] == (IBranch 8 (IBranch 2 (IBranch 1 ILeaf ILeaf) (IBranch 5 (IBranch 3 ILeaf (IBranch 4 ILeaf ILeaf)) ILeaf)) (IBranch 9 ILeaf ILeaf))

Arithmetic Expressons

data Expr = C Float | Add Expr2 Expr | Sub Expr2 Expr | Mul Expr2 Expr | Div Expr2 Expr Or, using infix constructor names: data Expr = C Float | Expr :+ Expr | Expr :- Expr | Expr :* Expr | Expr :/ Expr Infix constructors begin with a colon (:) , whereas ordinary constructor functions begin with an upper-case character.

Example

e1 = (C 10 :+ (C 8 :/ C 2)) :* (C 7 :- C 4) evaluate :: Expr -> Float evaluate (C x) = x evaluate (e1 :+ e2) = evaluate e1 + evaluate e evaluate (e1 :- e2) = evaluate e1 - evaluate e evaluate (e1 :* e2) = evaluate e1 * evaluate e evaluate (e1 :/ e2) = evaluate e1 / evaluate e Main> evaluate e