%July 1995, modified Feb 1996
%Paradigma: line bundle
%C.G. van der laan, cgl@rc.service.rug.nl
\input blue.tex 
\loadtocmacros
\def\xytxt#1{%Function: place text #1 at x, y
   \xy{\vbox to0pt{\vss
   \hbox to0pt{\strut#1\hss}\kern0pt}}}
\tolerance500\hbadness=499\hfuzz=8pt
%\parindent0pt %=1pc
\everyverbatim{\emc}

\bluetitle Paradigms: The winds and halfwinds

\bluesubtitle Details matter

\blueissue \maps{96}1

%\blueauthor Kees van der Laan

\bluepictures\pythtreepic\spiralpic\windpic\trimodelpic\tritreepic
   \bintreepic\rotatedbintreepic\chartpic

\beginscript
\everyverbatim{\unmc}
\bluehead BLUe's Design V

Hi folks. 
How to draw lines at {\oldstyle45}$^\circ$ in \TeX{} is exercised.
This simple-looking and innocent problem touches on fundamental issues, 
such as should we adhere to the Cartesian picture environment 
approach,\ftn{The \LaTeX{} picture environment for example.}
or do we need something else now and then?

This work emerged from studying Gurari's approach to graphics
incented by a suggestion in Jos Winnink's review, \maps{94}2.

The discussion below explains what has been used in BLUe's format system 
in the turtle graphics\ftn
{Knuth already used the `Turtle' idea in his dragon figures.
 For those interested in turtle graphics consult for example 
 S. Papert: Mindstorms.}
macros, especially in the coding of
\cs{N}, \cs{E}, \cs{S}, \cs{W},  
\cs{NE}, \cs{SE}, \cs{SW}, \cs{NW}, next to 
\cs{ESE}, and \cs{WSW}.
Examples are included which show what can be attained by these
basic functionalities.
It is also shown how the obtained pictures can be stored 
in the picture database |pic.dat| for reuse
with  different scaling or positioning.

In the Appendix some tree diversions have been given.

\bluehead What is the problem?

\TeX's \cs{hrule} and \cs{vrule} primitives are a gem and very powerful.
It would be nice to have similar primitives for any direction.
In absence of these we can use line pieces provided in fonts.
However, the latter approach suffers from the following drawbacks
\bitem for a few discrete directions only
\bitem line lengths are discrete too
\bitem line thickness is inflexible.
\smallbreak

\bluehead Why?

The need for general and flexible line elements arose when I faced the problem how
to draw classical fractals.\ftn{Inspired by Gurari's work on \TeX{} and graphics.}
The very least is the possibility to draw lines at {\oldstyle45}$^\circ$\Dash the halfwinds.

\blueexample Pythagorean tree

$$\pythtreepic$$
How to do this in \TeX\ftn
{For the coding of \cs{pythtree} see later. Or
 Macro writing\Dash with confidence.}?

Through the picture environment? 
Too clumsy, and cumbersome when changing the order, for example.

Through \PS?
This is possible, especially it provides for lines at all 
orientations.\ftn{J V Romanovsky has provided a concise \PS{} program.}

Through \MF? Definitely, see my \MF{} anthology.

However, why not go for how far we can get just by \TeX?

And what about the relevancy? I'm very pleased by the spin-off  how to
typeset binary trees or charts, even rotated without the use of \PS.
See the Appendix or the Publishing with \TeX{} user's guide.

What we need is not the Cartesian picture environment approach, 
but the good old penplotter \TeX niques, better known
in the pedagogical world as `Turtle graphics.'

\bluehead Turtle graphics

The basic idea is that the position of the turtle is maintained in 
the dimen variables \cs{x} and \cs{y},
with the reference point left invariant. 
The accompanying figure shows the effect of \cs{N1}: draw the line 
(\cs{x}, \cs{y})\dash(\cs{x}, \cs{y}+1), i.e.\ the turtle moves up.
After completion \cs{y} has been increased by \cs{unitlength}.

%Perhaps I better put it in pic.dat
\vskip2.5\bigskipamount
$$\unitlength.5ex
  \y=0pt\x-10\unitlength
  {\W{4}\E{40}}\S{4}%
  {\advance\y-2\unitlength\xytxt{ Ref.\ point}}%
  \N{20}\whiteE{20}\S4\whiteS4\xytxt{ (x,y)}$$
\vskip1.5\bigskipamount
The movements\Dash our first steps in the turtle graphics world\Dash 
can be achieved by the control sequences
\bitem \cs{N}, \cs{E}, \cs{S}, \cs{W} mean draw north, east, south and west, 
       similar for \cs{NE}, \cs{SE}, \cs{SW}, and \cs{NW}
\bitem \cs{whiteN}, \cs{whiteE}, \cs{whiteS}, \cs{whiteW} mean
      {\em  white}-draw north, east, south and west, i.e., the turtle just moves.\ftn{The
         halfwinds can be composed from the winds in this case.}
\smallbreak

\blueexample Spiral

To get the flavour another classical picture, which shows
that we don't have to worry about coordinates.
$$\spiralpic$$
The spiral is marked up essentially as follows\ftn{For the 
   placement within context details 
   consult \cs{spiralpic} in pic.dat.} 
\beginverbatim
\unitlength.5ex \k30 
\loop\E{\the\k}\advance\k-1
     \S{\the\k}\advance\k-1
     \W{\the\k}\advance\k-1
     \N{\the\k}\advance\k-1
\ifnum\k>4 \repeat
!endverbatim
\vskip-\medskipamount
\bluehead Basics

Essential for the understanding is familiarity with \TeX's
{\em boxes of size zero}, to know the effect of \cs{kern}-s and
\cs{h/vss}-s inside, and of combinations of these boxes.
Before reading on peruse the following.

\bluesubhead Kerns and stretchorshrink-s

within boxes of size zero.

\blueexample With and without boxes of size zero

\beginverbatim
\newdimen\x \x=4ex \newdimen\y \y=2ex
.\hbox to 0pt{\kern\x a\hss}. 
\kern10ex and \kern10ex
.\kern\x a.

and

.\hbox to0pt{\kern\x\vbox to0pt
   {\vss\hbox{a}\kern\y}\hss}.
\kern10ex and \kern10ex
.\kern\x\raise\y\hbox{a}.
!endverbatim
with result
%\newdimen\x \newdimen\y
\bgroup
\x4ex \y2ex
$$.\hbox to0pt{\kern\x a\hss}. 
\kern10ex and \kern10ex
.\kern\x a.$$
and
$$.\hbox to0pt{\kern\x\vbox to0pt
   {\vss\hbox{a}\kern\y}\hss}.
\kern10ex and \kern10ex
.\kern\x\raise\y\hbox{a}.$$
\egroup
By this mechanism we can move to
any point on the page and put there what we wish.
The essential issue of the box of zero width is that
the {\em reference point is left invariant.} 
It is the same before and after.

\bluesubhead Putting it together

In vertical mode the \cs{hbox}-es are aligned on the reference point, and 
when the heights and depths are zero the \cs{hbox}-es overprint,
and the order of specification is immaterial.

In (restricted) horizontal mode \cs{hbox}-es of width zero overprint
and can be given in any order, \TB{} {\oldstyle389}.

In math mode the zero size boxes overprint. 
In display math the invariant reference point is centered horizontally.

Coding in \TeX{} simple? Forget it!
So unusual, but \dots con\-sequent, utmost consequent, sigh. 

\bluesubhead Coding flashbacks

The codes which follow are the fifth or so version.
The zeroeth version was there to please me, excited me.
Recursive programming worked as I expected.
Then I had a version which served to try out as many examples as I could think of.
During the process of collecting examples I strove after consistency of the codes.
Why did I use integer arithmetic anyway? 
I had no good reason and I decided to
use dimensions and do the arithmetic through the dimensions, 
and widened en-passant the allowed arguments of the winds and
halfwinds to expand to `factor'-s.

And what about the context?

It seemed necessary to abstract from the multitude of boxes and to come up with
an enveloping hbox of width zero.

Finally, details were taken care of, such as how do the macros behave in the various
contexts.
Do the winds and halfwinds  behave the same?
It turned out that they did not. So I had to go back from where I started.
I considered the early versions as prototypes  and started 
anew guided by specs, 
with simplicity and robustness as yardsticks, 
stating precisely how the elements should behave.
By all means no sinecure, but IMHO with all respect,
a must in software engineering.\ftn{In fact I had this article nearly finished
   when I realized that the vertical rules induce a vbox of the size of the
   rules, while other directions when boxed yielded a box of size zero.
   Not nice and confusing, although for the article and in my hands it was ok.
   Nevertheless, I decided to go for a uniform approach. When boxed the dimensions
   of the box should be zero.}
Indeed, a superficially, simple problem.

Is that all? Essentially yes, but \dots details matter.
 
\bluehead The winds and halfwinds

The intuitive idea is to compose lines out of elements, from what I call atoms. 
I used  squares, casu quo rectangles, as atoms, and 
stacked these as follows.\ftn{In order to make it visible
  \cs{linethickness} has been set to 1ex. Experiments with bullets and \LaTeX's line fonts
   did not yield the pleasing exact results.}  

Essential  is how lines leave a mathematical point. 
The accompanying model picture  is obtained through the following macros.
\beginverbatim
\let\0\N \let\1\NE \let\2\E \let\3\SE 
\let\4\S \let\5\SW \let\6\W \let\7\NW 
\linethickness1ex
\setbox\hlfwndelm=\hbox{\vrule
   width\the\linethickness 
   height\the\linethickness depth0pt}
\unitlength10ex
\def\draw{\csname\the\dir\endcsname1}
$$\loop\ifnum\dir<8{\draw}\advance\dir1\repeat$$
!endverbatim

\thispicture{\linethickness1ex}
$$\windpic$$
Remarks. \cs{N} etc.\ draw in the implied directions with length of the argument
and \cs{unitlength} as scaling.



 
\bluehead Pondering aloud

Can we attain compatibility with \TeX's rules primitives?
I don't think so, alas. On the other hand we have to 
reconsider the leaving of the rules from a point. 

\bluesubhead Thickness

What is meant by thickness if we stack instead of tile?
What is the perceived blackness?

I presumed that stacking  with square elements
of size \cs{linethickness}${\times}$\cs{linethickness}\Dash 
   like in the example figure\Dash 
yields the same blackness as a rule of thickness \cs{linethickness}.\ftn{I also
   supplied macros\Dash tiled variants\Dash for  \cs{NW} and \cs{NE}.}

\bluesubhead Size

Usually the size along the x-axis must be provided.
I prefer to have the real size specified independently from the orientation
of the line.
However, the resulting size is not necessarily \#1${\times}$\cs{unitlength}.\ftn{To put 
   it in another way: the required length must be a multiple of the atom size.} 
In general the result differs at most by half the atom size, 
because it is composed of a multiple of the atomic element. 
We have to correct by $\sqrt2$ to compensate for the direction, when we
pace along one of the axes.
In the example the required length is {\oldstyle10}ex, 
with as result the  stacking of {\oldstyle7} atoms of size {\oldstyle1}ex.

For large lines one could think of combining \LaTeX's line{\oldstyle10}
line elements with the smaller elements. 
I refrained from that for two reasons.
Firstly, the \LaTeX's NW line element\Dash the |\char'145|\Dash 
did not fit exactly in the box, and 
secondly, the inflexibility of the thickness of the font elements.

\bluehead Design specs

With awareness of the above I specified the following for the
microscopic level\Dash the wind and halfwind commands proper\Dash
and for the macroscopic level\Dash the placement within context.

\bluesubhead Microscopics

The functionality is
to draw a line of the specified length in the direction as implied by the
control sequence name.
The general specifications read as follows.
\bitem as argument a `factor' is expected, to yield the required length
       \#1${\times}$\cs{unitlength}\ftn{The idea is that not only integer values
         can be specified but also decimal fractions.}
\bitem parameterization is by \cs{linethickness}
\bitem after drawing, the position of the turtle is at the end of each line,
       the reference point has been left invariant, and all
       the boxes have zero width, height and depth. 
\smallbreak
Extra for the four halfwinds the following.
\bitem parameterization is by \cs{hlfwndelm} and  \cs{linethickness}
       in there
\bitem draw a line of $\approx$ the specified size
\bitem the atoms are stacked diagonally, at the corners.
\smallbreak

\bluesubhead Macroscopics

The placement within context is the concern of the user.
Because of the zero dimensions of the boxes it is a nuisance
to skip or kern when using a picture to create the open space,
the niche for the picture.
Moreover, when the picture does not take dimensions we are in trouble
at page breaks. Therefore assistance is badly needed. 
The picture environment idea  combined with 
databases come to rescue. 
The {\it use\/} of prefab pictures has been simplified in this way, 
while there is flexibility via \cs{thispicture} to override the defaults.

Pictures can be stored in BLUe's format pic.dat database. 
Within each database element the default placement within context can be povided for.
Through the use of \cs{everypicture} and/or \cs{thispicture} the defaults can be 
overriden. 
This approach complies with the general principles adopted in BLUe's format system. 

\bluehead Coding the winds and halfwinds

In the fourth version, where emphasis was put on that all boxes have zero dimensions,
I also decided to separate to get at the (x, y)  position from what is put there.
This is much in the spirit of the second \cs{point} macro of \TB{} {\oldstyle389}, and
adheres to the {\it separation of concerns\/} adage.\ftn{IMHO it makes the code more
   trustworthy and circumvents pitfalls. 
   Especially, the mixing up of the kerns needed to get at (x, y) 
   with the kerns to position what has to be put at (x, y).}

The dimension variables \cs{x} and \cs{y} have after completion the values
of the coordinates of the end of the line.
\beginverbatim
\newbox\hlfwndelm
\newdimen\linethickness \linethickness1ex%.2pt
\newdimen\auxdim %linesize
%
\def\xy#1{%Function: place #1 at x, y
   \vbox to0pt{\kern-\y
   \hbox to0pt{\kern\x#1\hss}\vss}}
%
\def\xytxt#1{%Function: place text #1 at x, y
   \xy{\vbox to0pt{\vss
   \hbox to0pt{\strut#1\hss}\kern0pt}}}
%
\def\N#1{\xy{\kern-.5\linethickness
  \vbox to0pt{\vss
  \hrule height#1\unitlength
   width\linethickness}}%
\advance\y#1\unitlength}
%
\def\S#1{\advance\y-#1\unitlength{\N{#1}}}
%
\def\E#1{\xy{\vbox to0pt{\vss
   \hrule width#1\unitlength
          height\linethickness 
          depth0pt\vss
}}\advance\x#1\unitlength}
%
\def\W#1{\advance\x-#1\unitlength{\E{#1}}}
%
\def\NE#1{\auxdim#1\unitlength\correction
\loop\advance\auxdim-\wd\hlfwndelm
\ifdim\auxdim>-.5\wd\hlfwndelm
   \xy{\vbox to0pt{\vss\copy\hlfwndelm}}%
   \advance\x\wd\hlfwndelm
   \advance\y\ht\hlfwndelm 
\repeat}
%
\def\NW#1{\auxdim#1\unitlength\correction
\loop\advance\auxdim-\wd\hlfwndelm
\ifdim\auxdim>-.5\wd\hlfwndelm 
   \advance\x-\wd\hlfwndelm
   \xy{\vbox to0pt{\vss\copy\hlfwndelm}}%
   \advance\y\ht\hlfwndelm
\repeat}
%
\def\SW#1{\auxdim#1\unitlength\correction
\loop\advance\auxdim-\wd\hlfwndelm
\ifdim\auxdim>-.5\wd\hlfwndelm 
   \advance\x-\wd\hlfwndelm
   \advance\y-\ht\hlfwndelm
   \xy{\vbox to0pt{\vss\copy\hlfwndelm}}%
\repeat}
%
\def\SE#1{\auxdim#1\unitlength\correction
\loop\advance\auxdim-\wd\hlfwndelm
\ifdim\auxdim>-.5\wd\hlfwndelm 
   \advance\y-\ht\hlfwndelm
   \xy{\vbox to0pt{\vss\copy\hlfwndelm}}%
   \advance\x\wd\hlfwndelm
\repeat}
%For \NE and \NW the titled variants read
\def\NE#1{\auxdim#1\unitlength\correction
\loop\advance\auxdim-.5\wd\hlfwndelm
\ifdim\auxdim>-.25\wd\hlfwndelm
   \xy{\vbox to0pt{\vss\copy\hlfwndelm}}%
   \advance\x.5\wd\hlfwndelm
   \advance\y.5\ht\hlfwndelm 
\repeat}
%
\def\NW#1{\auxdim#1\unitlength\correction
\loop\advance\auxdim-.5\wd\hlfwndelm
\ifdim\auxdim>-.25\wd\hlfwndelm 
   \xy{\vbox to0pt{\vss
       \llap{\copy\hlfwndelm}}}%
   \advance\x-.5\wd\hlfwndelm
   \advance\y.5\ht\hlfwndelm
\repeat}
!endverbatim

\bluehead Coding a database element

As promised in the Macroscopics section the visibility of the picture\Dash
   that is the size\Dash
has to be added when the picture is stored in the database for (re)use.
In order to do so I added two extra layers. 
The structure of a pic.dat database element, and
a picture environment on top of for example
gkp's environment to allow user flexibility.

If we assume that the picture
proper has been created within gkp's
\beginverbatim
\def\gkpbeginpicture(#1,#2)(#3,#4)%
%#1, #2 xsize, ysize (dimensionless)
%#3, #4 xshift, yshift (dimensionless)
{...}  
\def\gkpendpicture{...} 
!endverbatim
then the flexibility layer reads
\thispicture{\emc}
\beginverbatim
\def\beginpicture{\bgroup
   \the\everypicture\the\thispicture
   \gkpbeginpicture(\the\xdim,\the\ydim)%
   (\the\xoffset,\the\yoffset)%
   <pictureproper>}  
\def\endpicture{...} 
!endverbatim
A pic.dat database entry consists of a quadruple:
\cs{lst} list element tag,
\cs{<name>pic} the name,
a group with defaults,\ftn{Demarcated by explicit braces.} and
the picture proper enclosed by
\cs{beginpicture} and \cs{endpicture}.
\beginverbatim
\lst\<name>pic{<defaults>}
\beginpicture
<pictureproper>
\endpicture
!endverbatim
 
\blueexample Coding the database element bintreepic

The \cs{bintreepic} element of the pic.dat database reads as follows.\ftn{Note
   that there is much flexibilty here. Also the \cs{bintree} macros can be 
   reused as such. \cs{bintreepic} comes down to an invoke of this macro
   with some flexibilty added.}
\beginverbatim
\lst\bintreepic{\bgroup
   \unitlength.5ex\kk32 
   \xoffset{-32} \yoffset{-2}%
   \xdim{66}\ydim{5}%
   \def\eertnib##1\bintree{\fi}
\beginpicture\bintree\endpicture\egroup}
%with in the kernel blue.tex
\def\bintree{\S1\ifnum\kk=2 \eertnib\fi
   \divide\kk2 {\W{\the\kk}\bintree}%
   \E{\the\kk}\bintree}
%and accounting for the leaves
\def\eertnib#1\bintree{\fi\global\advance\k1
   \whiteS1\xytxt{ \csname\the\k\endcsname}}
!endverbatim

\bluehead Coding recursion

Long I believed that tail recursion\Dash as used for example in the 
\cs{loop} \dots \cs{repeat}\Dash  is not sufficient. We also have
to cope with `Ackermann-like splittings' of the recursion. 
For the case of the Pythagorean tree for example, this means that at each recursion
level we have to take care of the left branch and the right branch, i.e.\ two invocations
must be done. 
This interferes with how to handle the termination of the recursion.
A problem? \TeX{} too difficult?
Hang on there are at least two elegant and general solutions.

\bluesubhead Recursion termination

\blueexample $n$!

As pedagogical example the calculation of $n$!
\beginverbatim
\n=...  %counter var
\nfac=1 %default
\def\fac{\ifnum\n=1 \caf\fi
   \multiply\nfac\n \advance\n-1 \fac}
\def\caf#1\fac{\fi}
!endverbatim
The termination is obtained because \cs{fac} is used as end terminator
in the definition of \cs{caf}. \cs{fac} is implicitly gobbled, and 
the process backtracks, resulting finally in termination.

\blueexample Pythagorean tree

Without explanation the Pythagorean tree recursive macro
is given below.\ftn
{Hopefully, it is self-explanatory, at least 
 with respect to the main flow. By the way in what order
 is the tree drawn?}
At each level we have two invocations\Dash the branching\Dash 
to take care of the left and right branch.
\beginverbatim
\def\pythtree{\ifnum\level=1 \eerthtyp\fi
    \advance\level-1 
    \multiply\kk23\divide\kk32%\sqrt2 
    {\leftbranch\draw\pythtree}%
     \rightbranch\draw\pythtree}
\def\eerthtyp#1\pythtree{\fi}
!endverbatim
No build up of \cs{fi}-s, no use of \cs{expandafter} nor \cs{let}. 
The \cs{let} mechanism has been used throughout in \TB.

\bluesubhead A la Knuth

In the coding of \cs{squine}\Dash a quadratic spline\Dash
Knuth showed how to terminate (tail) recursions in \TeX. 
For \cs{fac} this results in the following.
\beginverbatim
\def\fac{\ifnum\n>1 \expandafter\dofac\fi}
\def\dofac{\multiply\nfac\n 
   \advance\n-1 \fac}
!endverbatim

For the Pythagorean tree the result reads as follows.
\beginverbatim
\def\pythtree{\ifnum\level>1 
   \expandafter\dopyth\fi}
\def\dopyth{\advance\level-1 
    \multiply\kk23\divide\kk32
    {\leftbranch\draw\pythtree}%
     \rightbranch\draw\pythtree}
!endverbatim
Earlier I noticed the elegance of introducing an extra level, especially
to circumvent too many \cs{expandafter}-s in a row.

As an aside the use reads
\beginverbatim
\let\0\N \let\1\NE \let\2\E \let\3\SE 
\let\4\S \let\5\SW \let\6\W \let\7\NW
\def\leftbranch{\advance\dir7
 \ifnum\dir>7 \advance\dir-8 \fi}%mod 8
\def\rightbranch{\advance\dir1 
 \ifnum\dir>7 \advance\dir-8 \fi}%mod 8
\def\draw{\csname\the\dir\endcsname{\the\kk}}
\def\pythtree{\ifnum\level>1 
   \expandafter\dopyth\fi}
\def\dopyth{\advance\level-1 
    \multiply\kk23\divide\kk32
    {\leftbranch\draw\pythtree}%
     \rightbranch\draw\pythtree}
$$\unitlength0.1pt\kk128 %Size
                  \level5%Order
  \N{\the\kk}            %Trunk
  \pythtree$$
\tracingstats1
\bye
!endverbatim

\bluehead Trinaries

For {\oldstyle45}$^\circ$ lines I used square elements.
Why not use a rectangular element for {\oldstyle30}$^\circ$ lines
in conformance to the direction?

{\catcode`\^=7 \blueexample Lines at {\oldstyle30}$^\circ$

}
$$\trimodelpic$$
This model is obtained by
\beginverbatim
$$\x0pt\y0pt
  {\N{10}}{\ESE{10}}{\WSW{10}}$$
%with initializations
\linethickness1ex
\setbox\trielm=\hbox{\vrule 
   width1.74\linethickness 
   height\linethickness\relax}
%To account for element in 30 degrees direction
\unitlengthy\ht\trielm %default.2pt
\unitlengthx\wd\trielm %default.3482pt
\unitlength\unitlengthy%default.2pt
%and the macros
\def\WSW#1{\auxdim#1\unitlength\divide\auxdim2 
   \loop\advance\auxdim-\unitlengthy 
   \ifdim\auxdim>-.5\unitlengthy
      \advance\x-\unitlengthx
      \advance\y-\unitlengthy
      \xy{\vbox to0pt{\vss\copy\trielm}}%
   \repeat}
%
\def\ESE#1{\auxdim#1\unitlength\divide\auxdim2 
   \loop\advance\auxdim-\unitlengthy 
   \ifdim\auxdim>-.5\unitlengthy
      \advance\y-\unitlengthy
      \xy{\vbox to0pt{\vss\copy\trielm}}%
      \advance\x\unitlengthx
   \repeat}     
!endverbatim

\blueexample Trinary tree

$$\N{128}\tritreepic$$
\beginverbatim
$$\x0pt\y0pt\level6 \kk128\N{128}\tritree$$
%with trinary tree macro
\def\tritree{\ifnum1=\level \eertirt\fi
    \advance\level-1 \divide\kk2
    {\N{\the\kk}\tritree}%
    {\ESE{\the\kk}\tritree}%
     \WSW{\the\kk}\tritree}
\def\eertirt#1\tritree{\fi}
!endverbatim
Remark. The \cs{unitlength}-s are default 
        equal to the sides of the elementary rectangular block.
        The size of the tree can be controlled by \cs{kk}.

\bluehead Epilogue

The lines at {\oldstyle45}$^\circ$ have little compatibility
with \TeX's rules, alas, especially with non-neglible thickness. 
I was surprised to realize that \TeX's defaults for rules are 
not symmetric around their axes in relation to the reference point. 
\medskip\noindent
Have fun, and all the best.

\makesignature\let\makesignature\relax

\bluehead Appendix: Binary tree and chart

\blueexample Binary tree

\beginverbatim
\bluepictures\bintreepic\par
$$\bintreepic$$
!endverbatim
with result
$$\bintreepic$$

How can we add leaves\ftn{See the PWT user guide for the answer.}?

\bluesubhead Rotated tree

Once we understand turtle graphics, rotating a tree can be done easily 
by shifting the meaning of the directions, and adjusting the
positioning of the leaves.\ftn{A white lie. The tree is actually mirrored
   because I like the leaves to be numbered from the top. 
   In general we can rotate via \PS.}

\blueexample Rotated tree

\begindemo
\thispicture{\def\1{cgl}
  \def\2{PWT}\def\3{July}
  \def\4{1995}\def\5{\dots}
  \yoffset{-16}\ydim{28}}
$$\rotatedbintreepic$$
!yields
\thispicture{\def\1{cgl}\def\2{PWT}
  \def\3{July}\def\4{1995}
  \def\5{\dots}\def\6{}\def\7{}
  \yoffset{-16}\ydim{28}}
$$\rotatedbintreepic$$
\enddemo
\cs{bintree} and \cs{eertnib} come with blue.tex, and
\cs{rotatedtreepic} is included in pic.dat.
The \cs{rotatedtreepic} entry reads as follows.
\beginverbatim
\lst\rotatedbintreepic{%July 1995, cgl
\bgroup\unitlength1ex%
  \let\W\N \let\exchange\E 
  \let\E\S \let\S\exchange
  \def\1{x}\def\2{y}\def\3{a}\def\4{b}%
  \def\5{piet}\def\6{hans}\def\7{etc.}%
  \k0\kk16\xdim{10}\ydim{30}%
\beginpicture\bintree\endpicture\egroup
\thispicture{}}
!endverbatim 

\bluesubhead Chart

Through the \cs{bintree} macro we can also
obtain charts elegantly.

\blueexample Chart, \TB{} ex{\oldstyle22}.{\oldstyle14} {\oldstyle248}

$$\chartpic$$
obtained via
\beginverbatim
%labels in preorder (default in \chartpic)
\def\1{LMB, 1912}
\def\2{MJHB, 1882}\def\5{PAME, 1884}
\def\3{JHB, 1838} \def\4{MDB, 1840}
\def\6{EFE, 1845} \def\7{CLW, 1850}
\ekk8
\k0\unitlength2ex\x0pt\y0pt\kk8
\hbox{\bintree}
%with auxiliaries
\let\Eold\E
\def\E{\global\advance\k1
  \xytxt{ \csname\the\k\endcsname}\Eold}
!endverbatim
Remark. An aid in finding the numbers of the branches is
to delete \cs{csname} and \cs{endcsname} in \cs{E} .
The way of traversal at hand is called preorder.

When using \cs{chartpic} from |pic.dat| the texts along the branches
\Dash |\def\1{...} etc.|\Dash
have to be supplied within a \cs{thispicture},
to override the defaults.
\medbreak
\makesignature
\pasteuptoc
\endscript