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 — Optional challenge: Sorted?
Implement the function sorted?, which takes a tree and determines whether it is sorted in the sense that the numbers increase (or stay the same) in a inorder travsersal of the tree.
Your function should run in time proportional to the size of the tree, which rules out making a list of the tree numbers using append on recursive calls. One possible solution accumulates a compound value, and another possible solution accumulates information on the way down the tree and returns more than just success or failure.
