5. Functional data representation
5.1. Nats
Consider the following toy implementation of the type Nat
which encodes natural numbers.
trait Nat {} case object Zero extends Nat {} case class Succ(n: Nat) extends Nat {}
For instance, 3
will be encoded as the value: Succ(Succ(Succ(Zero)))
.
5.1.1. Write a function which implements addition over Nats:
def add(n: Nat, m: Nat): Nat = ???
5.1.2. Write a function which converts a Nat
to an Int
:
def toInt(n: Nat): Int = ???
5.1.3. Write a function which converts an Int
to a Nat
.
def fromInt(i: Int): Nat
5.2. Binary Search Trees
In a 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:
trait ITree {} case object Empty extends ITree case class INode(key: Int, left: ITree, right: ITree) extends ITree
5.2.1. Create the tree shown below:
val tree = ??? /* 5 / \ 2 7 / \ \ 1 3 9 */
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:
5 t.successor(2) = 5 / \ t.successor(5) = 6 2 7 t.successor(7) = 8 / \ 6 8
(!!) 5.2.11. Implement a method remove(k)
which removes element k
from the BST.