Lists and natural numbers are two classes of data whose description requires selfreferential data definitions. Both data definitions consist of two clauses; both have a single selfreference. Many interesting classes of data, however, require more complex definitions than that. Indeed, there is no end to the variations. It is therefore necessary to learn how to formulate data definitions on our own, starting with informal descriptions of information. Once we have those, we can just follow a slightly modified design recipe for selfreferential data definitions.
Medical researchers rely on family trees to do research on hereditary diseases. They may, for example, search a family tree for a certain eye color. Computers can help with these tasks, so it is natural to design representations of family trees and functions for processing them.
One way to maintain a family tree of a family is to add a node to the tree every time a child is born. From the node, we can draw connections to the node for the father and the one for the mother, which tells us how the people in the tree are related. For those people in the tree whose parents are unknown, we do not draw any connections. The result is a socalled ancestor family tree because, given any node in the tree, we can find the ancestors of that person if we follow the arrows but not the descendants.
As we record a family tree, we may also want to record certain pieces of information. The birth date, birth weight, the color of the eyes, and the color of the hair are the pieces of information that we care about. Others record different information.

See figure 35 for a drawing of an ancestor family tree. Adam is the child of Bettina and Carl; he has yellow eyes and was born in 1950. Similarly, Gustav is the child of Eva and Fred, has brown eyes, and was born in 1988. To represent a child in a family tree is to combine several pieces of information: information about the father, the mother, the name, the birth date, and eye color. This suggests that we define a new structure:
(definestruct child (father mother name date eyes))
The five fields of child
structures record the required
information, which suggests the following data definition:
AWhile this data definition is simple, it is unfortunately also useless. The definition refers to itself but, because it doesn't have any clauses, there is no way to create achild
is a structure:
where(makechild f m na da ec)
f
andm
arechild
structures;na
andec
are symbols; andda
is a number.
child
structure. If we tried to create a
child
structure, we would have to write
(makechild (makechild (makechild (makechild ... ))) ... ... ... ...)
without end. It is for this reason that we demand that all selfreferential data definitions consist of several clauses (for now) and that at least one of them does not refer to the data definition.
Let's postpone the data definition for a moment and study instead how we
can use child
structures to represent family trees. Suppose we
are about to add a child to an existing family tree, and furthermore
suppose that we already have representations for the parents. Then we can
just construct a new child
structure. For example, for Adam we
could create the following child
structure:
(makechild Carl Bettina 'Adam 1950 'yellow)
assuming Carl
and Bettina
stand for representations of
Adam's parents.
The problem is that we don't always know a person's parents. In the family
depicted in figure 35, we don't know Bettina's parents.
Yet, even if we don't know a person's father or mother, we must still use
some Scheme value for the two fields in a child
structure. We
could use all kinds of values to signal a lack of information (5
,
false
, or 'none
); here, we use empty
. For
example, to construct a child
structure for Bettina, we do the
following:
(makechild empty empty 'Bettina 1926 'green)
Of course, if only one of the two parents is missing, we fill just that
field with empty
.
Our analysis suggests that a child
node has the following data
definition:
A child node isThis definition is special in two regards. First, it is a selfreferential data definition involving structures. Second, the data definition mentions two alternatives for the first and second component. This violates our conventions concerning the shape of data definitions.(makechild f m na da ec)
where
f
andm
are either
empty
or
child
nodes;
na
andec
are symbols;
da
is a number.
We can avoid this problem by defining the collection of nodes in a family tree instead:
A familytreenode (short: ftn) is either
empty
; or
(makechild f m na da ec)
where f
and m
are ftns, na
and ec
are symbols, and da is a number.
This new definition satisfies our conventions. It consists of two clauses. One of the clauses is selfreferential, the other is not.
In contrast to previous data definitions involving structures, the
definition of ftn
is not a plain explanation of what kind of data
can show up in which field. Instead, it is multiclausal and
selfreferential. Considering that this is the first such data definition,
let us carefully translate the example from figure 35 and
thus reassure ourselves that the new class of data can represent the
information of interest.
The information for Carl is easy to translate into a ftn
:
(makechild empty empty 'Carl 1926 'green)
Bettina and Fred are represented with similar nodes. Accordingly, the node for Adam is created with
(makechild (makechild empty empty 'Carl 1926 'green) (makechild empty empty 'Bettina 1926 'green) 'Adam 1950 'yellow)
As the examples show, a simpleminded, nodebynode transliteration of
figure 35 requires numerous repetitions of data. For
example, if we constructed the child
structure for Dave like the
one for Adam, we would get
(makechild (makechild empty empty 'Carl 1926 'green) (makechild empty empty 'Bettina 1926 'green) 'Dave 1955 'black)
Hence it is a good idea to introduce a variable definition per node and to
use the variable thereafter. To make things easy, we use Carl
to
stand for the child
structure that describes Carl, and so on.
The complete transliteration of the family tree into Scheme can be found in
figure 36.
The structure definitions in figure 36 naturally correspond
to an image of deeply nested boxes. Each box has five compartments. The
first two contain boxes again, which in turn contain boxes in their first
two compartments, and so on. Thus, if we were to draw the structure
definitions for the family tree using nested boxes, we would quickly be
overwhelmed by the details of the picture. Furthermore, the picture would
copy certain portions of the tree just like our attempt to use
makechild
without variable definitions. For these reasons, it is
better to imagine the structures as boxes and arrows, as originally drawn
in figure 35. In general, a programmer must flexibly switch
back and forth between both of these graphical illustrations. For
extracting values from structures, the boxesinboxes image works best; for
finding our way around large collections of interconnected structures, the
boxesandarrows image works better.
Equipped with a firm understanding of the family tree representation, we can turn to the design of functions that consume family trees. Let us first look at a generic function of this kind:
;; funforftn : ftn > ???
(define (funforftn aftree) ...)
After all, we should be able to construct the template without considering the purpose of a function.
Since the data definition for ftn
s contains two clauses, the
template must consist of a condexpression with two clauses. The
first deals with empty
, the second with child
structures:
;;funforftn : ftn > ???
(define (funforftn aftree) (cond [(empty? aftree) ...] [else ;(child? aftree)
... ]))
Furthermore, for the first clause, the input is atomic so there is nothing further to be done. For the second clause, though, the input contains five pieces of information: two other family tree nodes, the person's name, birth date, and eye color:
;; funforftn : ftn > ???
(define (funforftn aftree)
(cond
[(empty? aftree) ...]
[else
... (funforftn (childfather aftree)) ...
... (funforftn (childmother aftree)) ...
... (childname aftree) ...
... (childdate aftree) ...
... (childeyes aftree) ...]))
We also apply funforftn
to the father
and
mother
fields because of the selfreferences in the second clause
of the data definition.
Let us now turn to a concrete example: blueeyedancestor?
, the
function that determines whether anyone in some given family tree has blue
eyes:
;;blueeyedancestor? : ftn > boolean
;; to determine whetheraftree
contains achild
;; structure with'blue
in theeyes
field (define (blueeyedancestor? aftree) ...)
Following our recipe, we first develop some examples. Consider the family tree node for Carl. He does not have blue eyes, and because he doesn't have any (known) ancestors in our family tree, the family tree represented by this node does not contain a person with blue eyes. In short,
(blueeyedancestor? Carl)
evaluates to false
. In contrast, the family tree represented by
Gustav
contains a node for Eva who does have blue eyes. Hence
(blueeyedancestor? Gustav)
evaluates to true
.
The function template is like that of funforftn
, except that we
use the name blueeyedancestor?
. As always, we use the template
to guide the function design. First we assume that (empty? aftree)
holds. In that case, the family tree is empty, and nobody has blue
eyes. Hence the answer must be false
.
The second clause of the template contains several expressions, which we must interpret:
(blueeyedancestor? (childfather aftree))
, which determines
whether someone in the father's ftn
has blue eyes;
(blueeyedancestor? (childmother aftree))
, which determines
whether someone in the mother's ftn
has blue eyes;
(childname aftree)
, which extracts the child
's name;
(childdate aftree)
, which extracts the
child
's date of birth; and
(childeyes aftree)
, which extracts the
child
's eye color.
It is now up to us to use these values properly. Clearly, if the
child
structure contains 'blue
in the eyes
field, the function's answer is true
. Otherwise, the function produces
true
if there is a blueeyed person in either the father's or the
mother's family tree. The rest of the data is useless.
Our discussion suggests that we formulate a conditional expression and that the first condition is
(symbol=? (childeyes aftree) 'blue)
The two recursions are the other two conditions. If either one produces
true
, the function produces true
. The else
clause
produces false
.
In summary, the answer in the second clause is the expression:
(cond [(symbol=? (childeyes aftree) 'blue) true] [(blueeyedancestor? (childfather aftree)) true] [(blueeyedancestor? (childmother aftree)) true] [else false])
The first definition in figure 37 pulls everything
together. The second definition shows how to formulate this
condexpression as an equivalent orexpression, testing
one condition after the next, until one of them is true
or all of
them have evaluated to false
.

The function blueeyedancestor?
is unusual in that it uses the
recursions as conditions in a condexpressions. To understand how
this works, let us evaluate an application of blueeyedancestor?
to Carl
by hand:
(blueeyedancestor? Carl) = (blueeyedancestor? (makechild empty empty 'Carl 1926 'green)) = (cond [(empty? (makechild empty empty 'Carl 1926 'green)) false] [else (cond [(symbol=? (childeyes (makechild empty empty 'Carl 1926 'green)) 'blue) true] [(blueeyedancestor? (childfather (makechild empty empty 'Carl 1926 'green))) true] [(blueeyedancestor? (childmother (makechild empty empty 'Carl 1926 'green))) true] [else false])]) = (cond [(symbol=? 'green 'blue) true] [(blueeyedancestor? empty) true] [(blueeyedancestor? empty) true] [else false]) = (cond [false true] [false true] [false true] [else false]) = false
The evaluation confirms that blueeyedancestor?
works properly
for Carl
, and it also illustrates how the function works.
Exercise 14.1.1.
The second definition of blueeyedancestor?
in
figure 37 uses an orexpression instead of a
nested conditional. Use a handevaluation to show that this definition
produces the same output for the inputs empty
and
Carl
.
Solution
(blueeyedancestor? empty)
evaluates to false
with a handevaluation.
Evaluate (blueeyedancestor? Gustav)
by hand and with
DrScheme. For the handevaluation, skip those steps in the evaluation that
concern extractions, comparisons, and conditions involving
empty?
. Also reuse established equations where possible,
especially the one above.
Solution
Exercise 14.1.3.
Develop countpersons
. The function consumes a family tree node
and produces the number of people in the corresponding family
tree.
Solution
Exercise 14.1.4.
Develop the function averageage
. It consumes a family tree node
and the current year. It produces the average age of all people in the
family tree.
Solution
Exercise 14.1.5.
Develop the function eyecolors
, which consumes a family tree node
and produces a list of all eye colors in the tree. An eye color may occur
more than once in the list.
Hint: Use the Scheme operation append
, which consumes two lists
and produces the concatenation of the two lists. For example:
(append (list 'a 'b 'c) (list 'd 'e)) = (list 'a 'b 'c 'd 'e)
We discuss the development of functions like append
in
section 17.
Solution
Exercise 14.1.6.
Suppose we need the function properblueeyedancestor?
. It is
like blueeyedancestor?
but responds with true
only when
some proper ancestor, not the given one, has blue eyes.
The contract for this new function is the same as for the old one:
;;properblueeyedancestor? : ftn > boolean
;; to determine whetheraftree
has a blueeyed ancestor (define (properblueeyedancestor? aftree) ...)
The results differ slightly.
To appreciate the difference, we need to look at Eva, who is blueeyed, but does not have a blueeyed ancestor. Hence
(blueeyedancestor? Eva)
is true
but
(properblueeyedancestor? Eva)
is false
. After all Eva
is not a proper ancestor of herself.
Suppose a friend sees the purpose statement and comes up with this solution:
(define (properblueeyedancestor? aftree) (cond [(empty? aftree) false] [else (or (properblueeyedancestor? (childfather aftree)) (properblueeyedancestor? (childmother aftree)))]))
What would be the result of (properblueeyedancestor? A)
for any ftn
A
?
Fix the friend's solution. Solution
Programmers often work with trees, though rarely with family trees. A particularly wellknown form of tree is the binary search tree. Many applications employ binary search trees to store and to retrieve information.
To be concrete, we discuss binary trees that manage information about
people. In this context, a binary tree is similar to a family tree but
instead of child
structures it contains node
s:
(definestruct node (ssn name left right))
Here we have decided to record the social security number, the name, and
two other trees. The latter are like the parent fields of family trees,
though the relationship between a node
and its left
and
right
trees is not based on family relationships.
The corresponding data definition is just like the one for family trees: A binarytree (short: BT) is either
false
; or
(makenode soc pn lft rgt)
where soc
is a number, pn
is a symbol,
and lft
and rgt
are BT
s.
The choice of false
to indicate lack of information is
arbitrary. We could have chosen empty
again, but false
is an
equally good and equally frequent choice that we should become familiar
with.
Here are two binary trees:
(makenode 15 'd false (makenode 24 'i false false))
(makenode 15 'd (makenode 87 'h false false) false)
Figure 38 shows how we should think about such
trees. The trees are drawn upside down, that is, with the root at the top
and the crown of the tree at the bottom. Each circle corresponds to a node,
labeled with the ssn
field of a corresponding node
structure. The trees omit false
.
Exercise 14.2.1.
Draw the two trees above in the manner of figure 38. Then
develop containsbt
. The function consumes a number and a
BT
and determines whether the number occurs in the
tree.
Solution
Exercise 14.2.2.
Develop searchbt
. The function consumes a number n
and a
BT
. If the tree contains a node
structure whose
soc
field is n
, the function produces the value of the
pn
field in that node. Otherwise, the function produces
false
.
Hint: Use containsbt
. Or, use boolean?
to find out
whether searchbt
was successfully used on a subtree. We will
discuss this second technique, called backtracking, in the intermezzo at
the end of this part.
Solution

Both trees in figure 38 are binary trees but they differ in a significant way. If we read the numbers in the two trees from left to right we obtain two sequences:
The sequence for tree A is sorted in ascending order, the one for B is not.
A binary tree that has an ordered sequence of information is a BINARY SEARCH TREE. Every binary search tree is a binary tree, but not every binary tree is a binary search tree. We say that the class of binary search trees is a PROPER SUBCLASS of that of binary trees, that is, a class that does not contain all binary trees. More concretely, we formulate a condition  or data invariant  that distinguishes a binary search tree from a binary tree:
The BST InvariantA binarysearchtree (short: BST) is a
BT
:
false
is always aBST
;
(makenode soc pn lft rgt)
is aBST
if
lft
andrgt
areBST
s,all
ssn
numbers inlft
are smaller thansoc
, andall
ssn
numbers inrgt
are larger thansoc
.
The second and third conditions are different from what we have
seen in previous data definitions. They place an additional and unusual
burden on the construction BST
s. We must inspect all numbers in
these trees and ensure that they are smaller (or larger) than
soc
.
Exercise 14.2.3.
Develop the function inorder
. It consumes a binary tree and
produces a list of all the ssn
numbers in the tree. The list
contains the numbers in the lefttoright order we have used above.
Hint: Use the Scheme operation append
, which concatenates
lists:
(append (list 1 2 3) (list 4) (list 5 6 7)) evaluates to (list 1 2 3 4 5 6 7)
What does inorder
produce for a binary search tree?
Solution
Looking for a specific node
in a BST
takes fewer steps
than looking for the same node
in a BT
. To find out
whether a BT
contains a node with a specific ssn
field, a
function may have to look at every node
of the tree. In contrast,
to inspect a binary search tree requires far fewer inspections than
that. Suppose we are given the BST
:
(makenode 66 'a L R)
If we are looking for 66
, we have found it. Now suppose we are
looking for 63
. Given the above node
, we can focus the
search on L
because all node
s with ssn
s
smaller than 66
are in L
. Similarly, if we were to look
for 99
, we would ignore L
and focus on R
because
all node
s with ssn
s larger than 66
are
in R
.
Exercise 14.2.4.
Develop searchbst
. The function consumes a number n
and
a BST
. If the tree contains a node
structure whose
soc
field is n
, the function produces the value of the
pn
field in that node. Otherwise, the function produces
false
. The function organization must exploit the BST
Invariant so that the function performs as few comparisons as
necessary. Compare searching in binary search trees with searching in
sorted lists (exercise 12.2.2).
Solution
Building a binary tree is easy; building a binary search tree is a
complicated, errorprone affair. To create a BT
we combine two
BT
s, an ssn
number and a name
with
makenode
. The result is, by definition, a BT
. To create
a BST
, this procedure fails because the result would typically not
be a BST
. For example, if one tree contains 3
and
5
, and the other one contains 2
and 6
, there is
no way to join these two BST
s into a single binary search tree.
We can overcome this problem in (at least) two ways. First, given a list of
numbers and symbols, we can determine by hand what the corresponding
BST
should look like and then use makenode
to build
it. Second, we can write a function that builds a BST
from the
list, one node
after another.
Exercise 14.2.5.
Develop the function createbst
. It consumes a BST
B
, a number N
, and a symbol S
. It produces a
BST
that is just like B
and that in place of one
false
subtree contains the node
structure
(makenode N S false false)
Test the function with (createbst false 66 'a)
; this should create a
single node
. Then show that the following holds:
(createbst (createbst false 66 'a) 53 'b) = (makenode 66 'a (makenode 53 'b false false) false)
Finally, create tree A from figure 38 using
createbst
.
Solution
Exercise 14.2.6.
Develop the function createbstfromlist
. It consumes a list of
numbers and names; it produces a BST
by repeatedly applying
createbst
.
The data definition for a list of numbers and names is as follows:
A listofnumber/name is either
empty
or
(cons (list ssn nom) lonn)
where ssn
is a number, nom
a symbol,
and lonn
is a listofnumber/name
.
Consider the following examples:


They are equivalent, although the left one is defined with the quote
abbreviation, the right one using list
. The left tree in
figure 38 is the result of using createbstfromlist
on this list.
Solution
The World Wide Web, or just ``the Web,'' has become the most interesting part of the Internet, a global network of computers. Roughly speaking, the Web is a collection of Web pages. Each Web page is a sequence of words, pictures, movies, audio messages, and many more things. Most important, Web pages also contain links to other Web pages.
A Web browser enables people to view Web pages. It presents a Web page as a sequence of words, images, and so on. Some of the words on a page may be underlined. Clicking on underlined words leads to a new Web page. Most modern browsers also provide a Web page composer. These are tools that help people create collections of Web pages. A composer can, among other things, search for words or replace one word with another. In short, Web pages are things that we should be able to represent on computers, and there are many functions that process Web pages.
To simplify our problem, we consider only Web pages of words and nested Web pages. One way of understanding such a page is as a sequence of words and Web pages. This informal description suggests a natural representation of Web pages as lists of symbols, which represent words, and Web pages, which represent nested Web pages. After all, we have emphasized before that a list may contain different kinds of things. Still, when we spell out this idea as a data definition, we get something rather unusual:
A Webpage (short: WP) is either
empty
;
(cons s wp)
where s
is a symbol and wp
is a Web page; or
(cons ewp wp)
where both ewp
and wp
are Web pages.
This data definition differs from that of a list of symbols in
that it has three clauses instead of two and that it has three
selfreferences instead of one. Of these selfreferences, the one at the
beginning of a cons
tructed list is the most unusual. We refer
to such Web pages as immediately embedded
Web pages.
Because the data definition is unusual, we construct some examples of Web pages before we continue. Here is a plain page:
'(The TeachScheme! Project aims to improve the problemsolving and organization skills of high school students. It provides software and lecture notes as well as exercises and solutions for teachers.)
It contains nothing but words. Here is a complex page:
'(The TeachScheme Web Page Here you can find: (LectureNotes for Teachers) (Guidance for (DrScheme: a Scheme programming environment)) (Exercise Sets) (Solutions for Exercises) For further information: write to [email protected])
The immediately embedded pages start with parentheses and the symbols
'LectureNotes
, 'Guidance
, 'Exercises
, and
'Solutions
. The second embedded Web page contains another embedded
page, which starts with the word 'DrScheme
. We say this page is
embedded
with respect to the entire page.
Let's develop the function size
, which consumes a Web page and
produces the number of words that it and all of its embedded pages contain:
;;size : WP > number
;; to count the number of symbols that occur inawp
(define (size awp) ...)
The two Web pages above suggest two good examples, but they are too complex. Here are three examples, one per subclass of data:
(= (size empty) 0)
(= (size (cons 'One empty)) 1)
(= (size (cons (cons 'One empty) empty)) 1)
The first two examples are obvious. The third one deserves a short explanation. It is a Web page that contains one immediately embedded Web page, and nothing else. The embedded Web page is the one of the second example, and it contains the one and only symbol of the third example.
To develop the template for size
, let's carefully step through the
design recipe. The shape of the data definition suggests that we need three
cond
clauses: one for the empty
page, one for a page that
starts with a symbol, and one for a page that starts with an embedded Web
page. While the first condition is the familiar test for empty
,
the second and third need closer inspection because both clauses in the
data definition use cons
, and a simple cons?
won't
distinguish between the two forms of data.
If the page is not empty
, it is certainly cons
tructed,
and the distinguishing feature is the first item on the list. In other
words, the second condition must use a predicate that tests the first item
on awp
:
;; size : WP > number
;; to count the number of symbols that occur in awp
(define (size awp)
(cond
[(empty? awp) ...]
[(symbol? (first awp)) ... (first awp) ... (size (rest awp)) ...]
[else ... (size (first awp)) ... (size (rest awp)) ...]))
The rest of the template is as usual. The second and third cond
clauses contain selector expressions for the first item and the rest of the
list. Because (rest awp)
is always a Web page and because
(first awp)
is one in the third case, we also add a recursive
call to size for these selector expressions.
Using the examples and the template, we are ready to design size
:
see figure 39. The differences between the definition and the
template are minimal, which shows again how much of a function we can design
by merely thinking systematically about the data definition for its inputs.

Exercise 14.3.1.
Briefly explain how to define size
using its template and the
examples. Test size
using the examples from above.
Exercise 14.3.2.
Develop the function occurs1
. The function consumes a Web page and a
symbol. It produces the number of times the symbol occurs in the Web page,
ignoring the nested Web pages.
Develop the function occurs2
. It is like occurs1
, but
counts all occurrences of the symbol, including in embedded Web
pages.
Solution
Exercise 14.3.3.
Develop the function replace
. The function consumes two symbols,
new
and old
, and a Web page, awp
. It
produces a page that is structurally identical to awp
but with
all occurrences of old
replaced by new
.
Solution
Exercise 14.3.4.
People do not like deep Web trees because they require too many page
switches to reach useful information. For that reason a Web page designer
may also want to measure the depth of a page. A page containing only
symbols has depth 0
. A page with an immediately embedded page has
the depth of the embedded page plus 1
. If a page has several
immediately embedded Web pages, its depth is the maximum of the depths of
embedded Web pages plus 1
. Develop depth
, which consumes
a Web page and computes its depth.
Solution
DrScheme is itself a program that consists of several parts. One function checks whether the definitions and expressions we wrote down are grammatical Scheme expressions. Another one evaluates Scheme expressions. With what we have learned in this section, we can now develop simple versions of these functions.
Our first task is to agree on a data representation for Scheme programs. In other words, we must figure out how to represent a Scheme expression as a piece of Scheme data. This sounds unusual, but it is not difficult. Suppose we just want to represent numbers, variables, additions, and multiplications for a start. Clearly, numbers can stand for numbers and symbols for variables. Additions and multiplications, however, call for a class of compound data because they consist of an operator and two subexpressions.
A straightforward way to represent additions and multiplications is to use two structures: one for additions and another one for multiplications. Here are the structure definitions:
(definestruct add (left right)) (definestruct mul (left right))
Each structure has two components. One represents the left expression and the other one the right expression of the operation.
Scheme expression  representation of Scheme expression 

3  3 
x  'x 
(* 3 10)  (makemul 3 10) 
(+ (* 3 3) (* 4 4))  (makeadd (makemul 3 3) (makemul 4 4)) 
(+ (* x x) (* y y))  (makeadd (makemul 'x 'x) (makemul 'y 'y)) 
(* 1/2 (* 3 3))  (makemul 1/2 (makemul 3 3)) 
Let's look at some examples:
These examples cover all cases: numbers, variables, simple expressions, and nested expressions.
Exercise 14.4.1. Provide a data definition for the representation of Scheme expressions. Then translate the following expressions into representations:
(+ 10 10)
(+ (* 20 3) 33)
(* 3.14 (* r r))
(+ (* 9/5 c) 32)
(+ (* 3.14 (* o o)) (* 3.14 (* i i)))
Solution
A Scheme evaluator is a function that consumes a representation of a Scheme
expression and produces its value. For example, the expression 3
has the value 3
, (+ 3 5)
has the value 8
,
(+ (* 3 3) (* 4 4))
has the value 25
, etc. Since
we are ignoring definitions for now, an expression that contains a
variable, for example, (+ 3 x)
, does not have a value; after all,
we do not know what the variable stands for. In other words, our Scheme
evaluator should be applied only to representations of expressions that do
not contain variables. We say such expressions are numeric.
Exercise 14.4.2.
Develop the function numeric?
, which consumes (the representation
of) a Scheme expression and determines whether it is
numeric.
Solution
Exercise 14.4.3.
Provide a data definition for numeric expressions. Develop the function
evaluateexpression
. The function consumes (the representation of)
a numeric Scheme expression and computes its value. When the function is
tested, modify it so it consumes all kinds of Scheme expressions; the
revised version raises an error when it encounters a
variable.
Solution
Exercise 14.4.4.
When people evaluate an application (f a)
they substitute
a
for f
's parameter in f
's body. More generally,
when people evaluate expressions with variables, they substitute the
variables with values.
Develop the function subst
. The function consumes (the
representation of) a variable (V
), a number (N
), and
(the representation of) a Scheme expression. It produces a structurally
equivalent expression in which all occurrences of V
are
substituted by N
.
Solution