Description
Objectives
• Mergesort and binary trees
Activities
1. mergesort works recursively by splitting its input in halves, sorting each half recursively using
mergesort, and using merge to combine the two sorted halves. In English: if the list has zero or
one element,you are done, otherwise split the list into 2 equal-sized sublists, sort each sublist,
and merge them into a sorted result. Notice that unlike Quicksort there is no pivot: we split
the list so the two parts are equal size (within 1) without regard to the values in the list.
You wrote the merge function in Problem Set 6. Feel free to use that code, or this:
( define ( merge la lb )
( cond (( null ? la ) lb )
(( null ? lb ) la )
(( < ( car la )( car lb ))
( cons ( car la )( merge ( cdr la ) lb )))
( else
( cons ( car lb )( merge la ( cdr lb ))))))
(a) Write a function (split l) that takes a list of numbers and partitions it into two equalsized (within one) lists, and returns a list whose first element is the first list and whose
second element is the second list.
You can use the built-in length function if you would like, but it should not be called more
than once (in total) during split.
(b) Using split and your merge function from last week (or the one listed above), write a
function (mergesort l) as described above that takes a list of numbers and returns it
sorted in ascending order.
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2. This question involves working with binary trees (trees where a node may 0, 1, or 2 children).
Recall the conventions we have adopted in class for maintaining trees. We represent the empty
tree with the empty list (); a nonempty tree is represented as a list of three objects
(value left-subtree right-subtree)
where value is the value stored at the root of the tree, and left-subtree and right-subtree
are the two subtrees. We introduced some standardized functions for maintaining and accessing
this structure, which you should use in your solutions below.
( define ( make-tree value left right )
;; produces a tree with value at the root , and
;; left and right as its subtrees .
( list value left right ))
( define ( value tree )
( car tree ))
( define ( left tree )
( cadr tree ))
( define ( right tree )
( caddr tree ))
Notably, you will need to use these functions to produce some test trees to test the following
functions. For example, to produce the simple tree in Figure 1, you could use the following
code:
( define testtree ( make-tree 1
( make-tree 3
( make-tree 7 ’() ’())
( make-tree 9 ’() ’()))
( make-tree 5 ’() ’())))
You will probably want to make some more complex trees than that, but you get the idea.
Finally, for all of these, figure out how to define the function recursively, as that should map
fairly directly to code.
(a) A non-empty tree is made up of nodes: the root, and all of the nodes in its subtrees. Define
a Scheme procedure (tree-node-count t) that calculates the number of nodes in tree t.
It should also work for empty trees. (tree-node-count testtree) would be 5.
(b) Each node in a tree has a value. Define a Scheme procedure (tree-node-sum t) that
calculates the sum of the values of the nodes in nodes in tree t. It should work for any
binary tree whose node values are numbers, as well as for empty trees. (tree-node-sum
testtree) would be 25.
(c) The height of a node in a tree is the length of the longest path from that node to a leaf –
we can consider the height of a tree to be the height of its root. The height of the empty
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1
3 5
7 9
Figure 1: Tree testtree produced by code in question 2.
tree is undefined, the height of a node without children (leaf) is 0, otherwise the height of
a node is one more than the maximum height of the trees rooted at its children.
Define a Scheme procedure (tree-height t) that calculates the height of a non-empty
tree t. testtree has height 2.
(d) Define a Scheme procedure, named (tree-map f t), which takes two parameters, a function, f and a tree t, and is analogous to the map function for lists. Namely, it returns a
new tree with a topology identical to that of t but where each node in the new tree has the
value f(v), where v is the value at the corresponding position in t. For example, if the input
tree is testtree shown in Figure 1 then the tree returned by (tree-map (lambda(x)(* 3
x)) testtree) is shown in Figure 2.
3
3
9 15
21 27
Figure 2: Tree resulting from mapping (lambda(x)(* x 3)) over testtree.
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