Laboratory Exercises For Computer Science 153

An Introduction To Trees

Before Class: Read section 7.2 in the text.

Goals: This lab reviews elements of the tree data structure.

Discussion: A general tree is defined recursively as follows:

A null tree (or null list) is a tree.
If I is a data element and if T₁, T₂, T₃, ..., T_n are n trees, then

is also a tree.

As noted in the text, I is called the root or root node, and each T_i is called a subtree of the tree. Nodes that have only null subtrees are called leaves or leaf nodes.

In Scheme, such a tree may be represented as the list:

(I  (T₁)  (T₂)  (T₃)  ...  (T_n))

Since this definition is recursive, it may be applied multiple times to construct more complex trees, such as the one shown below: A Simple Tree

In this example, e, k, l, m, n, h, o, p, and q are leaves (or trees with null subtrees); a is the root of the overall tree. f is the root of a tree with k and l as subtrees, etc. Similarly, b is the root of a tree with two subtrees, one containing e and one containing f, k, and l. In Scheme/list notation, the tree with b as root would be notated

(b (e) (f (k) (l)))

Write the subtrees with roots c and d in Scheme/list notation.
Answer exercises 7-9 and 7-10 from the text.

In a file accompanying the textbook, the authors give the following definitions:

(define t 
  '(1 (2 (3))
      (4 (7)
         (5 (6)))
      (10 (11 (12) (13))
          (19))))
(define u 
  '(1 (2 (3))
      (4 (7)
         (5 (6)))
      (19 (11 (12) (13))
          (10))))

Draw the tree structures corresponding to these list structures.

What are the leaves of each tree?

Section 7.2 discusses the general format for processing within a tree structure. Review the algorithms and code given.

Program 7-6 presents a procedure to print the nodes in a tree, using a letrec statement. Write a paragraph explaining how that program works.
Rewrite Program 7-6 using a named-let expression.
Test your code with trees t and u above.
Could Program 7-6 be rewritten without recursion -- replacing the recursion in the letrec statement with a do statement?
Assuming that all nodes in a non-empty tree contain numbers, write a procedure to find the sum of these numbers.
Test your code with trees t and u above.
Again, assuming that all nodes in a non-empty tree contain numbers, write a procedure to find the maximum of these values.
Test your code with trees t and u above.

This document is available on the World Wide Web as

http://www.math.grin.edu/~walker/courses/153/lab-intro-trees.html

created April 18, 1998
last revised April 18, 1998