Datatypes, creating functions

In this homework you will gain skills by working on two different tasks.

Creating and defining functions over user defined datatypes
Creating functions in Haskell

Part 1. Creating and defining functions over user defined data.

Study the notes we covered in class. This includes the presentation and the examples we defined in class. Then answer the following questions by writing a Haskell program.

A useful data type is called Maybe

data Maybe a = Nothing | Just a

It has two constructors Nothing and Just. It is often used when some input or result is optional or may or may not be possible. Pattern matching will have two clauses. For example a function that determines if the Just constructor is used might look like:

isJust Nothing = False
isJust (Just x) = True

The maybe type is often used to indicate if something is possible. For example when given a list of pairs, one might search for the first pair where the first part of the pair is equal to some key, and return Just the second part of the pair if a matching pair is found. Note that sometimes no possible matching key is found, so that is when the Nothing constructor is returned.

(1) Write the lookup function with contract given below.

find :: key -> [(key,value)] -> Maybe value

Before you begin be sure you.

Create a bunch of examples using HUnit assertions
Figure out how many cases there are. Ask yourself do you case over the list or the Maybe type?
How will you structure the comparison of the key. Will you use guarded clauses? if expressions?

(2) Write a data definition for a binary tree (i.e. branching trees have two branches). There should be two non-branching constructors Empty which is a constant, and One which stores a single element of any type. The Branch constructor should have exactly two sub trees, and nothing else.

data BinTree a = ...

(3) Using the constructor functions of BinTree: create 3 different trees of type BinTree Int. Recall the tree constructors are: Empty, One, and Branch. Be sure your trees vary in the number of Int elements are stored in your three example trees.

tree1,tree2,tree3:: BinTree Int
tree1 = undefined
tree2 = undefined
tree3 = undefined

(4) Write two functions over BinTrees.

flatten:: (BinTree a) -> [a]

The idea is to create a list of all the elements of type a in BinTree a.

and

CountTree:: (BinTree Int) -> Int

The idea is to sum up all the elemenets of type Int in a BintTree Int. If no elements appear return zero.

Before you begin be sure you.

Create a bunch of examples using HUnit assertions
Figure out how many cases there are.
Think about how will you structure the patterns?
What design recipes will you use?
- Generic program design recipe.
- Recursive program design recipe.
- Array processing design recipes.

Part 2. Creating functions in Haskell

Study the crib sheet 12 ways to create a function. Recall the source for this cribsheet can be found on the Haskell Resouce Sheet.

(5) Write the function plus5 three different ways. where (plus5 3) --> 8, (plus5 0) --> 5, etc.

```
 plus5a :: Int -> Int 
```
Using an anonymous function (lambda expression).
```
 plus5b :: Int -> Int 
```
Using a section.
```
 plus5c :: Int -> Int 
```
Using top level equations.

(6) Write functions smallest and largest (do not use the library functions min or max). Use guarded clauses for smallest and an if expression for largest.

smallest :: Ord t => t -> t -> t
largest  :: Ord t => t -> t -> t

(7) Write the function ncopies. Use a where clause and a comprehension.

ncopies:: Int -> a -> [a]

What to turn in.

Get a copy of this worksheet (by right-clicking and the downloading) and fill in the missing properties for each of the sorts.

Your name (as the author of the program) on the very first line
Each of the properties filled in the apprpriate blank.
The whole think uploaded to blackboard
Due before class on Tuesday, October 20, 2009.
You may cut and paste out the part of this document not related to your answers (such as the list of properties and the instructions of what to turn in).

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