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--- fp:lab05 [2022/03/28 13:35]
pdmatei
+++ fp:lab05 [2022/03/31 08:53] (current)
pdmatei
@@ Line 1: / Line 1: @@
 ===== 5. Functional data representation =====
-===== 5.1. Nats ====
+==== 5.1. Nats ===
+Consider the following toy implementation of the type ''Nat'' which encodes natural numbers.
 <code scala>
@@ Line 8: / Line 10: @@
 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 = ???
+/*
+      /   \
+     7
+    / \     \
+  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>
+             t.successor(2) = 5
+   / \            t.successor(5) = 6
+   7           t.successor(7) = 8
+     / \
+   8
+</code>
+** (!!) 5.2.11.** Implement a method ''remove(k)'' which removes element ''k'' from the BST.