===== 5. Functional data representation =====

==== 5.1. Nats ===

Consider the following toy implementation of the type ''Nat'' which encodes natural numbers.

<code scala>
trait Nat {}
case object Zero extends Nat {}
case class Succ(n: Nat) extends Nat {}
</code>

For instance, ''3'' will be encoded as the value: ''Succ(Succ(Succ(Zero)))''.

**5.1.1.** Write a function which implements addition over Nats:
<code scala>
def add(n: Nat, m: Nat): Nat = ???
</code>

**5.1.2.** Write a function which converts a ''Nat'' to an ''Int'':
<code scala>
def toInt(n: Nat): Int = ???
</code>

**5.1.3.** Write a function which converts an ''Int'' to a ''Nat''. 
<code scala>
def fromInt(i: Int): Nat
</code>

==== 5.2. Binary Search Trees ===

In a [[https://en.wikipedia.org/wiki/Binary_search_tree| binary search tree (BST)]], the key of the current node, is always:
  * **smaller** or equal than **all** keys in the **right** sub-tree.
  * **larger** or equal than **all** keys in the **left** sub-tree. 

Consider a binary search tree with keys as integers, encoded as follows:
<code scala>
trait ITree {}
case object Empty extends ITree 
case class INode(key: Int, left: ITree, right: ITree) extends ITree 
</code>

**5.2.1.** Create the tree shown below:
<code scala>
val tree = ???
/*
        5
      /   \
     2     7
    / \     \ 
   1  3      9 
*/
</code>

**5.2.2.** Implement the method ''size'' which determines the number of non-empty nodes from the BST.

**5.2.3.** Define the method ''contains'', which checks if a given integer is a member of the BST.

**5.2.4.** Implement the method ''ins'' which inserts a new integer in the BST. **Note:** the insertion must return a new BST (the //binary search tree// property mentioned above must hold after insertion).

**5.2.5.** Implement a method ''flatten'' which converts a BST into a list of integers. You must carefully choose the flattening method in such a way as to obtain **a sorted list** from the BST. Hint: you may use the list concatenation operator '':::'' (triple colons; example usage: ''List(1,2,3):::List(4,5)''.

**5.2.6.** Implement a method ''depth'' which returns the maximal depth of a BST. Hint: use the method: ''_.max(_)''.

**(!) 5.2.8.** Implement a method ''minimum'' which returns the smallest integer from a BST. (If the tree is empty, we return -1). Hint: use the example above, to guide your implementation.

**5.2.9.** Implement a similar method ''maximum''. 

**(!) 5.2.10.** Implement a method ''successor(k)'' which returns **the smallest** integer from the BST, which is **larger** than ''k''. Use the following examples for your implementation:
<code>
    5             t.successor(2) = 5                      
   / \            t.successor(5) = 6
  2   7           t.successor(7) = 8
     / \          
    6   8
</code>

** (!!) 5.2.11.** Implement a method ''remove(k)'' which removes element ''k'' from the BST.