Due: Wednesday Jan 18, 2023 11:59pm

Difficulty: Medium

The following Tree datatype implements a binary tree with a number in each node and leaf:

(define-type Tree

(leaf [val : Number])

(node [val : Number]

      [left : Tree]

      [right : Tree]))

Part 1 — Sum

Implement a sum function that takes a tree and returns the sum of the numbers in the tree.

Example: (sum (node 5 (leaf 6) (leaf 7))) should produce 18.

Part 2 — Negate

Implement the function negate, which takes a tree and returns a tree that has the same shape, but with all the numbers negated.

Example: (negate (node 5 (leaf 6) (leaf 7))) should produce (node -5 (leaf -6) (leaf -7)).

Part 3 — Contains?

Implement the function contains?, which takes a tree and a number and returns #t if the number is in the tree, #f otherwise.

Example: (contains? (node 5 (leaf 6) (leaf 7)) 6) should produce #t.

The second argument to the contains? function is “along for the ride.”

Part 4 — Big Leaves?

Implement the function big-leaves?, which takes a tree and returns #t if every leaf is bigger than the sum of numbers in the path of nodes from the root that reaches the leaf.

Examples: (big-leaves? (node 5 (leaf 6) (leaf 7))) should produce #t, while (big-leaves? (node 5 (node 2 (leaf 8) (leaf 6)) (leaf 7))) should produce #f (since 6 is smaller than 5 plus 2).

The big-leaves? function should be a thin wrapper on another function, perhaps bigger-leaves?, that accumulates a sum of node values.

Part 5 — Positive Trees?

Implement the function positive-trees?, which takes a list of trees and returns #t if every member of the list is a positive tree, where a positive tree is one whose numbers sum to a positive value.

Hint 1: This function takes a list, not a tree, so don’t try to use the template for a tree function.

Hint 2: Use your sum function as a helper.

Hint 3: (positive-trees? empty) should produce #t, because there’s no tree in the empty list whose numbers sum to 0 or less.

More examples:

(test (positive-trees? (cons (leaf 6)

                           empty))

    #t)

(test (positive-trees? (cons (leaf -6)

                           empty))

    #f)

(test (positive-trees? (cons (node 1 (leaf 6) (leaf -6))

                           empty))

    #t)

(test (positive-trees? (cons (node 1 (leaf 6) (leaf -6))

                           (cons (node 0 (leaf 0) (leaf 1))

                                  empty)))

    #t)

(test (positive-trees? (cons (node -1 (leaf 6) (leaf -6))

                           (cons (node 0 (leaf 0) (leaf 1))

                                  empty)))

    #f)