Homework 5 CIS 252 (Spring 2009) — Introduction to Computer Science
Coverage
This assignment focuses on material in Chapter 14 of Haskell: The Craft of Func-
tional Programming.
Logistics
This homework is officially due in the bin in CST 3-212 by noon on Thursday,
March 5. However, it comes with an automatic extension: anything submitted
by noon on Friday, March 6 will be accepted as being on time.
You may work singly or in pairs on this assignment.
As always, you’re expected to follow the design recipe discussed in lecture.
Turn in a hard copy of your source code and a transcript demonstrating con-
vincingly that your code is correct. Also, submit your code (but not transcript)
electronically.
Background: Binary Trees
Trees are ubiquitous data structures in Computer Science: they’re used to order
data (in what are known as search trees), to represent syntactic objects for mini-
languages, and to provide a hierarchical structure on (say) file directories. This
write-up should include all the definitions you need for this homework. If you
want more information, however, check out:
http://en.wikipedia.org/wiki/Binary_tree
Abinary tree is a tree in which every node has at most two children, and the
children are ordered (that is, there is a left child and a right child). We can define
(polymorphic) binary trees in Haskell using the following type definition:
data BTree a = Empty
| BNode a (BTree a) (BTree a)
deriving (Show)
According to this definition, each node has a label: every label in a given tree
has the same type, just as every element in a list has the same type.
As an example, (Node 7 Empty Empty) is the BTree Int representation of the
tree on the left below (the filled-in circles represent empty trees):
7
BBBBy y
8
@@
13
BB
20
BBBB
12
BBBB
y yy
y y
The BTree Int representation of the tree on the right is:
BNode 8 (BNode 13 Empty (BNode 12 Empty Empty)) (BNode 20 Empty
Empty).
Here are some common definitions and examples:
•Aleaf is a node that has no children. The nodes labeled 7, 12, and 20 above
are leaves; the nodes labeled 8 and 13 are not leaves.
•The left child (respectively, right child) of the node labeled by 8 is the
node labeled by 13 (respectively, 20). The node labeled by 13 has a right
child, but not a left child.
The left subtree (respectively, right subtree) of a node nis the tree rooted
by n’s left child (respectively, right child). That is, a child is a single node;
a subtree is a node, plus all of its descendants (ie, its children’s children,
and their children, and so on).1
•The depth of a node is its distance to the root of the tree (i.e., the topmost
node). Thus the depth of the node labeled 7 is 0, while the depth of the
node labeled 12 is 2.
•The height of a tree is the length of path from the root to the deepest leaf
of the tree. The height of the tree rooted at 8 is 2.
•A binary tree is full if it has no nodes with only one child (i.e., all of its
nodes are either leaves or have two children).
1The distinction between children and subtrees is blurred by our representation of binary trees; you don’t need to worry about the distinction for this assignment.
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