On first blush, that appears to be astonishingly ... astonishing. Um. That's not good English is it. I'll try again.

After scant consideration, that appears to be remarkably ... remarkable. D'oh!

I will adapt it to produce my preferred, revised format -- contrast original format with my revised, preferred format so that I can compare like-for-like with my final version.

Then I'll get back to you.

Meantime, that's impressive!

Weirdly, I especially like the comments. If that sounds strange, trust me, it is even stranger, given that I (generally) hate comments.

For that you require a left/right flag to indicate which side of a subtree you are in. The logic is probably even simpler:

use strict;
use warnings;

use constant { LEFT => 0, RIGHT => 1 };

sub prepare {
  my( $subtree, $lr, $level, $col, $graph ) = @_;
  if( ref($subtree) ) {
    my $lcol = prepare( $subtree->[LEFT ], LEFT,  $level+1, $col, $gra
+ph );
    $$col+=2;
    my $rcol = prepare( $subtree->[RIGHT], RIGHT, $level+1, $col, $gra
+ph );
    $graph->[$level]->[$lcol] = '/';
    $graph->[$level-1]->[$_] = '_' for $lcol+1..$rcol-1;
    $graph->[$level]->[$rcol] = '\\';    
    return $lr == LEFT ? $rcol : $lcol;  # intentionally the other way
+ round !!
  } else {
    $graph->[$level]->[$$col] = $subtree;      
    return $$col;
  }
}

my $root = do {
  my $r;
  my @a = ( 'a'..'z', 1..9, 'A'..'Z' );
  $r = int( rand $#a ), splice @a, $r, 2, [ @a[ $r, $r+1 ]] while @a >
+ 1;
  $a[0];
};

my @graph;
my $col = 0;
prepare( $root, LEFT, 1, \$col, \@graph ); # need to start with $level
+ = 1 !
for my $row ( @graph ) {
  print $_ // ' ' for @$row;
  print "\n";
}
[download]

and it looks like this:

                                                 _____________
                                   _____________/             \_______
+__________
                             _____/             \_________   _/       
+          \___________________________
               _____________/     \_______     _/         \ / \    ___
+__________/                           \_________
     _________/             \    _/       \_  / \___      4 5 6  _/   
+          \_______                 ___/         \_
   _/         \_____        o  _/ \_     _/ \ x /   \_          / \_  
+       ___/       \_             _/   \_       _/ \
 _/ \_     ___/     \_        / \ / \_  / \ w   y  _/ \_        7 / \_
+   ___/   \___    / \_          / \  _/ \_    / \ Z
/ \ / \  _/   \_   _/ \_      p q r / \ u v       / \ / \         8 / 
+\ /   \_  /   \_  J / \___      Q R / \ / \_  X Y
a b c d / \_  / \ / \ / \_          s t           z 1 2 3           9 
+A B  _/ \ F  _/ \   K /   \_        S T U / \
        e / \ h i j k l / \                                           
+    / \ E   / \ I     L  _/ \_            V W
          f g           m n                                           
+    C D     G H         / \ / \
                                                                      
+                        M N O P
[download]

UPDATE:

Replacing the 2 in line $$col+=2; with a larger number makes the tree wider which might be useful for readability.

Replacing the line $graph->[$level]->[$$col] = $subtree; with $graph->[$level++]->[$$col] = $_ for split //, $subtree; will print leaf values vertically.

Very nice. Beautiful even. Makes mine look positively primitive :)

---

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Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.

"Science is about questioning the status quo. Questioning authority".

In the absence of evidence, opinion is indistinguishable from prejudice.