These problems must be done in scheme using Dr. Racket.

1. [3 marks] Write a function called tree-reduce for trees that is analogous to the reduce function for flat lists (see Section 4.2). This function should take an operator, an initial value, and a tree as arguments and return the result of combining all nodes of the tree using the given operator. For simplicity you may assume only commutative operators are supplied. E.g.:

    (tree-reduce + 0 '(1 (2 3)((4 5) 6 (7)((8 (9)) 10)))  55 

2. [3 marks] Write a function height that takes as argument an arbitrarily deeply nested list (ie a tree) and returns the maximum depth of any item in any of its sublists; the depth of an object in a list is the number of cars that can be applied to it, not necessarily in a row... (You may wish to use the built-in max function for this). E.g.:

    (height 'a)  0 (height '(a))  1 (height '(a (b) c))  2 (height '(((((a(((b)))))))))  8 

3. [3 marks] Write a function flatten-tree that takes a tree as an argument and returns a non-nested list of the terminal values of the tree in their original relative order. E.g.:

    (flatten-tree '(1 (2 3) ((4 5 6 (7)))(((8 (9))))))  (1 2 3 4 5 6 7 8 9) 

4. [6 marks] Write a function (tree-merge T1 T2) that takes two trees as arguments and merges them according to the following rules: E.g.:
   • Merging two trees is done by recursing through their structure and merging their subtrees.
     • The root of T1 is merged with the root of T2
     • The first child of T1 is merged with the first child of T2
     • Second with second... etc, etc, etc, ...
   • Merging two leaf nodes is done by multiplying their values (you may assume they are numbers).
   • Merging a leaf node with a subtree is done by scaling the subtree by the value of the leaf.
   • Merging a subtree with an empty tree, is simply the non-empty subtree.