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%! TeX program = lualatex
\documentclass[a4paper]{article}
% packages
\usepackage{microtype} % Slightly tweak font spacing for aesthetics
\usepackage[english]{babel} % Language hyphenation and typographical rules
\usepackage[final, colorlinks = false, urlcolor = cyan]{hyperref}
\usepackage{changepage} % adjust margins on the fly
\usepackage{fontspec}
\usepackage{minted}
\usemintedstyle{algol_nu}
\usepackage{xcolor}
\usepackage{pgfplots}
\pgfplotsset{width=\textwidth,compat=1.9}
\usepackage{caption}
\newenvironment{code}{\captionsetup{type=listing}}{}
\captionsetup[listing]{skip=0pt}
\setlength{\abovecaptionskip}{5pt}
\setlength{\belowcaptionskip}{5pt}
\usepackage[yyyymmdd]{datetime}
\renewcommand{\dateseparator}{--}
\setmainfont{EB Garamond}
\setmonofont[Scale=MatchLowercase]{Deja Vu Sans Mono}
\usepackage{titlesec}
% \titleformat{\section}{\LARGE\bfseries}{}{}{}[\titlerule]
% \titleformat{\subsection}{\Large\bfseries}{}{0em}{}
% \titlespacing{\subsection}{0em}{-0.7em}{0em}
%
% \titleformat{\subsubsection}{\large\bfseries}{}{0em}{$\bullet$ }
% \titlespacing{\subsubsection}{1em}{-0.7em}{0em}
% margins
\addtolength{\hoffset}{-2.25cm}
\addtolength{\textwidth}{4.5cm}
\addtolength{\voffset}{-3.25cm}
\addtolength{\textheight}{5cm}
\setlength{\parskip}{0pt}
\setlength{\parindent}{0in}
% \setcounter{secnumdepth}{0}
\begin{document}
\hrule \medskip
\begin{minipage}{0.295\textwidth}
\raggedright
\footnotesize
Name: Andrew Hayes \\
E-mail: \href{mailto://a.hayes18@universityofgalway.ie}{\texttt{a.hayes18@universityofgalway.ie}} \hfill\\
ID: 21321503 \hfill
\end{minipage}
\begin{minipage}{0.4\textwidth}
\centering
\vspace{0.4em}
\Large
\textbf{CT331} \\
\end{minipage}
\begin{minipage}{0.295\textwidth}
\raggedleft
\today
\end{minipage}
\medskip\hrule
\begin{center}
\normalsize
Assignment 2: Functional Programming with Scheme
\end{center}
\hrule
\section{Question 1}
\subsection{Part (A): Code}
\begin{code}
\inputminted[texcl, mathescape, breaklines, frame=single]{racket}{../code/assignment_q1.rkt}
\caption{\texttt{assignment\_q1.rkt}}
\end{code}
\subsection{Part (B): Comments}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{./images/question1.png}
\caption{Output of \texttt{assignment\_q1.rkt}}
\end{figure}
Comments on each line of output:
\begin{enumerate}
\item The \mintinline{racket}{cons} function creates a \mintinline{racket}{cons} pair, which is not always a ``list''.
A list is a \mintinline{racket}{cons} pair in which the second element is another itself another list or is \mintinline{racket}{empty}.
When a \mintinline{racket}{cons} pair that is not a list is printed, its elements are delimited by a ``\verb|.|'', as can be seen from the first line of
output.
\item The second section of code produces a list of three numbers using only the \mintinline{racket}{cons} function:
first we create a one-element list with \mintinline{racket}{(cons 3 empty)}, then we create a two-element list by making a \mintinline{racket}{cons} pair
of \mintinline{racket}{2} and the already-created one-element list, and finally we create a three-element list by making a \mintinline{racket}{cons} pair
of \mintinline{racket}{1} and the two-element list.
This could of course be achieved far more simply by just using \mintinline{racket}{(cons 1 '(2 3))} or even justs \mintinline{racket}{'(1 2 3)} but I
felt that this would be against the spirit of the exercise.
\item To create a nested list using only \mintinline{racket}{cons} in the third section of code, we make the \mintinline{racket}{'(1 2 3)} as previously,
\mintinline{racket}{cons} it with \mintinline{racket}{empty} to make a nested list, and then \mintinline{racket}{cons} it with \mintinline{racket}{0}, and
\mintinline{racket}{cons} that with a string literal.
\item Like \mintinline{racket}{cons}, \mintinline{racket}{list} can take either atomics or lists as arguments.
To create the list using only the \mintinline{racket}{list} function, we can simply make a list of \mintinline{racket}{(list 1 2 3)}, and then create a
list consisting of \mintinline{racket}{"a string"}, \mintinline{racket}{0}, \& the aforementioned list.
This is much simpler than using \mintinline{racket}{cons} because \mintinline{racket}{list} can take as many arguments as we want, while
\mintinline{racket}{cons} can only take two arguments.
\item Although I opted not to make use of the ``\mintinline{racket}{'}'' operator to create lists for the previous exercises, I make use of it here as there is
no other way to create a list using only \mintinline{racket}{append} and nothing else, as \mintinline{racket}{append} only accepts lists as arguments.
We make a list consisting of only one element (\mintinline{racket}{"a string"}), another list consisting of only one element (\mintinline{racket}{0}),
and finally a list consisting of three elements \mintinline{racket}{'(1 2 3)} and append them into one to create the desired list.
\end{enumerate}
\section{Question 2}
\begin{code}
\inputminted[breaklines, frame=single]{racket}{../code/assignment_q2.rkt}
\caption{\texttt{assignment\_q2.rkt}}
\end{code}
\section{Question 3}
\begin{code}
\inputminted[breaklines, frame=single]{racket}{../code/assignment_q3.rkt}
\caption{\texttt{assignment\_q3.rkt}}
\end{code}
It is worth noting here that the function \mintinline{racket}{sort_ascending_last} operates in a manner that may be undesirable.
The function determines whether the two numbers passed to it as arguments were passed in strictly ascending order based on the final
digit, i.e. it returns \mintinline{racket}{#t} if the final digit of the first argument is less than the final digit of the second
argument, and \mintinline{racket}{#f} otherwise.
Because this function considers only the final digit of the numbers, it considers two numbers to be equal if they share a final digit,
e.g. it considers the numbers \mintinline{racket}{99} \& \mintinline{racket}{9} to be the same.
Therefore, if one attempts to insert those two values into the binary search tree using this function as the ``sorter'', only the
former value will get inserted, as binary search trees do not allow duplicate values, and the \mintinline{racket}{sort_ascending_last} function
considers those two values to be equal.
However, I thought it would be incorrect for the function to consider the $n-1$\textsuperscript{th} digits in the case of the $n$\textsuperscript{th} digits being identical, as
a) the assignment did not ask for that and b) that would really just be no different to sorting the numbers in ascending order.
\end{document}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{o}{\PYGZsh{}}\PYG{n+nv}{lang}\PYG{+w}{ }\PYG{n+nv}{racket}
\PYG{c+c1}{;; a cons pair of two numbers}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{p}{)}
\PYG{c+c1}{;; a list of 3 numbers using only the cons function}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nv}{empty}\PYG{p}{)))}\PYG{+w}{ }
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers using only the cons function}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+s}{\PYGZdq{}a string\PYGZdq{}}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nv}{empty}\PYG{p}{)))}\PYG{+w}{ }\PYG{n+nv}{empty}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the list function}
\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+s}{\PYGZdq{}a string\PYGZdq{}}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{))}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the append function}
\PYG{p}{(}\PYG{n+nb}{append}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+s}{\PYGZdq{}a string\PYGZdq{}}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+m+mi}{0}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{((}\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{)))}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{err}{\PYGZsh{}}\PYG{n+nv}{lang}\PYG{+w}{ }\PYG{n+nv}{racket}
\PYG{c+c1}{;; a cons pair of two numbers}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{p}{)}
\PYG{c+c1}{;; a list of 3 numbers using only the cons function}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nv}{empty}\PYG{p}{)))}\PYG{+w}{ }
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers using only the cons function}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+s}{\PYGZdq{}a string\PYGZdq{}}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nv}{empty}\PYG{p}{)))}\PYG{+w}{ }\PYG{n+nv}{empty}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the list function}
\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+s}{\PYGZdq{}a string\PYGZdq{}}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{))}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the append function}
\PYG{p}{(}\PYG{n+nb}{append}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+s}{\PYGZdq{}a string\PYGZdq{}}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+m+mi}{0}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{((}\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{)))}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{k+kn}{\PYGZsh{}lang }\PYG{n+nn}{racket}
\PYG{c+c1}{;; a cons pair of two numbers}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{p}{)}
\PYG{c+c1}{;; a list of 3 numbers using only the cons function}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{)))}\PYG{+w}{ }
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers using only the cons function}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+s+s2}{\PYGZdq{}a string\PYGZdq{}}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{)))}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the list function}
\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+s+s2}{\PYGZdq{}a string\PYGZdq{}}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{))}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the append function}
\PYG{p}{(}\PYG{n+nb}{append}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+s+s2}{\PYGZdq{}a string\PYGZdq{}}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+m+mi}{0}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{p}{(}\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{)))}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{n+nb}{cons}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{l+m+mi}{2}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
\PYG{k+kn}{\PYGZsh{}lang }\PYG{n+nn}{racket}
\PYG{c+c1}{;; a cons pair of two numbers}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{p}{)}
\PYG{c+c1}{;; a list of 3 numbers using only the cons function}
\PYG{c+c1}{;; this could be more easily done using the single quote `'` (i.e., `'(1 2 3)`) but i don't use it as it seemed against the spirit of the question}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{)))}\PYG{+w}{ }
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers using only the cons function}
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+s+s2}{\PYGZdq{}a string\PYGZdq{}}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{)))}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the list function}
\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+s+s2}{\PYGZdq{}a string\PYGZdq{}}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{))}
\PYG{c+c1}{;; a list containing a string, a number, and a nested list of three numbers, using only the append function}
\PYG{c+c1}{;; using `'` as the arguments of the `append` function must be themselves lists}
\PYG{p}{(}\PYG{n+nb}{append}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+s+s2}{\PYGZdq{}a string\PYGZdq{}}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+m+mi}{0}\PYG{p}{)}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{p}{(}\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{)))}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{n+no}{\PYGZsh{}f}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{n+nb}{append}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{n+nb}{list}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
\PYG{k+kn}{\PYGZsh{}lang }\PYG{n+nn}{racket}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{ins\PYGZus{}beg}\PYG{p}{)}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{ins\PYGZus{}end}\PYG{p}{)}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{count\PYGZus{}top\PYGZus{}level}\PYG{p}{)}
\PYG{c+c1}{;; function to insert an element at the beginning of a list}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{ins\PYGZus{}beg}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; assuming that the second element is always a list}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; function to insert an element at the end of a list}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{ins\PYGZus{}end}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; making el into a list if it isn't already}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{append}\PYG{+w}{ }\PYG{n}{lst}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{))}
\PYG{p}{)}
\PYG{c+c1}{;; function to count the number of top-level items in a list}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}top\PYGZus{}level}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{c+c1}{;; return 0 if we've reached the end of the list}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{+}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}top\PYGZus{}level}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)))}\PYG{+w}{ }\PYG{c+c1}{;; return 1 plus the count_top_level of the second element of the cons pair (the rest of the list)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{k+kn}{\PYGZsh{}lang }\PYG{n+nn}{racket}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{ins\PYGZus{}beg}\PYG{p}{)}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{ins\PYGZus{}end}\PYG{p}{)}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{count\PYGZus{}top\PYGZus{}level}\PYG{p}{)}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{count\PYGZus{}instances}\PYG{p}{)}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{count\PYGZus{}instances\PYGZus{}tr}\PYG{p}{)}
\PYG{p}{(}\PYG{k}{provide}\PYG{+w}{ }\PYG{n}{count\PYGZus{}instances\PYGZus{}deep}\PYG{p}{)}
\PYG{c+c1}{;; function to insert an element at the beginning of a list}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{ins\PYGZus{}beg}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; assuming that the second element is always a list}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; function to insert an element at the end of a list}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{ins\PYGZus{}end}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; making el into a list if it isn\PYGZsq{}t already}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{append}\PYG{+w}{ }\PYG{n}{lst}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{n}{el}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{))}
\PYG{p}{)}
\PYG{c+c1}{;; function to count the number of top\PYGZhy{}level items in a list}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}top\PYGZus{}level}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{c+c1}{;; return 0 if we\PYGZsq{}ve reached the end of the list}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{+}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}top\PYGZus{}level}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)))}\PYG{+w}{ }\PYG{c+c1}{;; return 1 plus the count\PYGZus{}top\PYGZus{}level of the second element of the cons pair (the rest of the list)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; non\PYGZhy{}tail recursive function to count the number of times a given item occurs in a list (assuming items are atomic)}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{c+c1}{;; return 0 if at the end of the list}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{+}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{equal?}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{))}
\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{c+c1}{;; if the item is equal to the first element of the list, add 1}
\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{+w}{ }\PYG{c+c1}{;; if the item is not equal to the first element of the list, add 0}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{))}\PYG{+w}{ }\PYG{c+c1}{;; recurse with the remainder of the list}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; helper function for count\PYGZus{}instances\PYGZus{}tr}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}tr\PYGZus{}helper}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{lst}\PYG{+w}{ }\PYG{n}{cnt}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{cond}
\PYG{+w}{ }\PYG{c+c1}{;; return the count if the end of the list is reached (0 for empty list)}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{n}{cnt}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the first element of the list is equal to the item }
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{eq?}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{item}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; recurse with the remainder of the list and an incremented count}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}tr\PYGZus{}helper}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{+}\PYG{+w}{ }\PYG{n}{cnt}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the first element of the list is not equal to the item}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{else}
\PYG{+w}{ }\PYG{c+c1}{;; recurse with the remainder of the list and an unchanged count}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}tr\PYGZus{}helper}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{cnt}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; tail recursive function to count the number of times a given item occurs in a list (assuming items are atomic)}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}tr}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; calling helper function with the list and the count so far (0)}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}tr\PYGZus{}helper}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{lst}\PYG{+w}{ }\PYG{l+m+mi}{0}\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; function to count the number of times an item occurs in a list and its sub\PYGZhy{}lists}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}deep}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{cond}
\PYG{+w}{ }\PYG{c+c1}{;; return nothing if we\PYGZsq{}ve reached the end of the list}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{l+m+mi}{0}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the first item is a list, recurse through the first element and then the rest and return the sum of the two results}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{pair?}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{+}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}deep}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{))}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}deep}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the first element is equal to the item, add 1 to the count and recurse with the rest of the list}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{eq?}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{))}\PYG{+w}{ }\PYG{c+c1}{; If the first element is equal to the item, increment count}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{+}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}deep}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; else if the first element is not equal to the item, recurse with the rest of the list}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{else}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{count\PYGZus{}instances\PYGZus{}deep}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{))}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{k+kn}{\PYGZsh{}lang }\PYG{n+nn}{racket}
\PYG{c+c1}{;; function to display the contents of a binary search tree in sorted order}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{display\PYGZus{}contents}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{cond}
\PYG{+w}{ }\PYG{c+c1}{;; if the binary search tree is null, print an empty string (nothing)}
\PYG{+w}{ }\PYG{p}{[}\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{display}\PYG{+w}{ }\PYG{l+s+s2}{\PYGZdq{}\PYGZdq{}}\PYG{p}{)]}
\PYG{+w}{ }\PYG{c+c1}{;; if the binary search tree has nodes}
\PYG{+w}{ }\PYG{p}{[}\PYG{k}{else}\PYG{+w}{ }
\PYG{+w}{ }\PYG{c+c1}{;; display the contents of the left sub\PYGZhy{}tree of the current node}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{display\PYGZus{}contents}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cadr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{c+c1}{;; display the current node}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{display}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{newline}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; display the contents of the right sub\PYGZhy{}tree of the current node}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{display\PYGZus{}contents}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{caddr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{]}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; function to search a tree and tell whether a given item is presesnt in a given tree}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{search\PYGZus{}tree}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{cond}
\PYG{+w}{ }\PYG{c+c1}{;; return false if we\PYGZsq{}ve reached the end of the tree without finding a match}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}f}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; return true if the current node is equal to the item}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{equal?}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}t}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; else return whether the item was found in the left sub\PYGZhy{}tree or the right sub\PYGZhy{}tree}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{else}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{or}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{search\PYGZus{}tree}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cadr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}\PYG{+w}{ }\PYG{c+c1}{;; search left sub\PYGZhy{}tree}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{search\PYGZus{}tree}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{caddr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}\PYG{+w}{ }\PYG{c+c1}{;; search right sub\PYGZhy{}tree}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; function to insert an item into a binary search tree}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{cond}
\PYG{+w}{ }\PYG{c+c1}{;; if there are no nodes in the tree, create a new tree with the item as the root}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{()}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{())}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the item is less than the current node, insert it into the left\PYGZhy{}hand side of the tree}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{\PYGZlt{}}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{c+c1}{;; create new bst with same root node, same right\PYGZhy{}hand side, but a left\PYGZhy{}hand side that has had the item inserted}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cadr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{caddr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the item is greater than the current node, insert it into the right\PYGZhy{}hand side of the tree}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{\PYGZgt{}}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{c+c1}{;; create new bst with same root node, same left\PYGZhy{}hand side, but a right\PYGZhy{}hand side that has had the item inserted}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cadr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{caddr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; else the item is equal to the current node, so do nothing}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{else}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; function to insert a list of items into a binary search tree}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}list}\PYG{+w}{ }\PYG{n}{lst}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the list is null, just return the bst with no changes}
\PYG{+w}{ }\PYG{n}{bst}
\PYG{+w}{ }\PYG{c+c1}{;; otherwise, recurse with the remainder of the list and the binary tree produced by inserting the first item of the list into bst}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}list}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{)}\PYG{+w}{ }
\PYG{p}{)}
\PYG{c+c1}{;; tree\PYGZhy{}sort function}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{tree\PYGZus{}sort}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; inserting the list into a tree structure to sort it and then displaying the contents of that tree }
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{display\PYGZus{}contents}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}list}\PYG{+w}{ }\PYG{n}{lst}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{()))}
\PYG{p}{)}
\PYG{c+c1}{;; function to insert an item into a binary search tree based off a sorting function}
\PYG{c+c1}{;; the sorting function should return accept two items and arguments, and return true if they were passed in order, and false otherwise or if they are equal}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item\PYGZus{}custom}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{n}{bst}\PYG{+w}{ }\PYG{n}{sorter}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{cond}
\PYG{+w}{ }\PYG{c+c1}{;; if there are no nodes in the tree, create a new tree with the item as the root}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{()}\PYG{+w}{ }\PYG{o}{\PYGZsq{}}\PYG{p}{())}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the item is goes before the current node, insert it into the left\PYGZhy{}hand side of the tree}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n}{sorter}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{c+c1}{;; create new bst with same root node, same right\PYGZhy{}hand side, but a left\PYGZhy{}hand side that has had the item inserted}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item\PYGZus{}custom}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cadr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{sorter}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{caddr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the item goes after the current node, insert it into the right\PYGZhy{}hand side of the tree}
\PYG{+w}{ }\PYG{p}{(}\PYG{p}{(}\PYG{n}{sorter}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{item}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; create new bst with same root node, same left\PYGZhy{}hand side, but a right\PYGZhy{}hand side that has had the item inserted}
\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cadr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item\PYGZus{}custom}\PYG{+w}{ }\PYG{n}{item}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{caddr}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{sorter}\PYG{p}{))}
\PYG{+w}{ }\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; else the item is equal to the current node, so do nothing}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{else}\PYG{+w}{ }\PYG{n}{bst}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; sorter function which states whether the two arguments were supplied in strictly ascending order (i.e., if item == item2, return false)}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{sort\PYGZus{}ascending}\PYG{+w}{ }\PYG{n}{item1}\PYG{+w}{ }\PYG{n}{item2}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{\PYGZlt{}}\PYG{+w}{ }\PYG{n}{item1}\PYG{+w}{ }\PYG{n}{item2}\PYG{p}{)}
\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}t}
\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}f}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; sorter function which states whether the two arguments were supplied in strictly descending order (i.e., if item == item2, return false)}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{sort\PYGZus{}descending}\PYG{+w}{ }\PYG{n}{item1}\PYG{+w}{ }\PYG{n}{item2}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{\PYGZgt{}}\PYG{+w}{ }\PYG{n}{item1}\PYG{+w}{ }\PYG{n}{item2}\PYG{p}{)}
\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}t}
\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}f}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; sorter function which states whether the two arguments were supplied in strictly ascending order based on the final digit (i.e., if item == item2, return false)}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{sort\PYGZus{}ascending\PYGZus{}last}\PYG{+w}{ }\PYG{n}{item1}\PYG{+w}{ }\PYG{n}{item2}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{\PYGZlt{}}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{modulo}\PYG{+w}{ }\PYG{n}{item1}\PYG{+w}{ }\PYG{l+m+mi}{10}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{modulo}\PYG{+w}{ }\PYG{n}{item2}\PYG{+w}{ }\PYG{l+m+mi}{10}\PYG{p}{))}
\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}t}
\PYG{+w}{ }\PYG{n+no}{\PYGZsh{}f}
\PYG{+w}{ }\PYG{p}{)}
\PYG{p}{)}
\PYG{c+c1}{;; function to insert a list of items into a binary search tree in the order determined by a sorting function}
\PYG{p}{(}\PYG{k}{define}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}list\PYGZus{}custom}\PYG{+w}{ }\PYG{n}{lst}\PYG{+w}{ }\PYG{n}{bst}\PYG{+w}{ }\PYG{n}{sorter}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{(}\PYG{k}{if}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{null?}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}
\PYG{+w}{ }\PYG{c+c1}{;; if the list is null, just return the bst with no changes}
\PYG{+w}{ }\PYG{n}{bst}
\PYG{+w}{ }\PYG{c+c1}{;; otherwise, recurse with the remainder of the list and the binary tree produced by inserting the first item of the list into bst}
\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}list\PYGZus{}custom}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{cdr}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}\PYG{+w}{ }\PYG{p}{(}\PYG{n}{insert\PYGZus{}item\PYGZus{}custom}\PYG{+w}{ }\PYG{p}{(}\PYG{n+nb}{car}\PYG{+w}{ }\PYG{n}{lst}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{bst}\PYG{+w}{ }\PYG{n}{sorter}\PYG{p}{)}\PYG{+w}{ }\PYG{n}{sorter}\PYG{p}{)}
\PYG{+w}{ }\PYG{p}{)}\PYG{+w}{ }
\PYG{p}{)}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{n}{sort\PYGZus{}ascending\PYGZus{}last}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{l+m+mi}{9}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{n+nb}{empty}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{l+m+mi}{0}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{l+s+s2}{\PYGZdq{}a string\PYGZdq{}}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{p}{(}\PYG{n+nb}{list}\PYG{+w}{ }\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{)}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{l+m+mi}{1}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{l+m+mi}{99}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{o}{\PYGZsq{}}\PYG{p}{(}\PYG{l+m+mi}{1}\PYG{+w}{ }\PYG{l+m+mi}{2}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{p}{)}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{p}{(}\PYG{n+nb}{cons}\PYG{+w}{ }\PYG{l+m+mi}{3}\PYG{+w}{ }\PYG{n+nb}{empty}\PYG{p}{)}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{n+no}{\PYGZsh{}t}
\end{Verbatim}

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\begin{Verbatim}[commandchars=\\\{\}]
\PYG{o}{\PYGZsq{}}
\end{Verbatim}

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\def\PYG@reset{\let\PYG@it=\relax \let\PYG@bf=\relax%
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\def\PYG@tok#1{\csname PYG@tok@#1\endcsname}
\def\PYG@toks#1+{\ifx\relax#1\empty\else%
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\def\PYG@do#1{\PYG@bc{\PYG@tc{\PYG@ul{%
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\def\PYGZob{\char`\{}
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% for compatibility with earlier versions
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