It looks like the container of daughters is an anonymous array, so given your 'original data' and 'desired re-ordered data' it all amounts to re-ordering the daughters array by their name. Maybe you want a method 'walk_down_sorted' or monkey-patch the module in walk_down amounting to:

--- /usr/share/perl5/Tree/DAG_Node.pm    2018-02-13 23:16:43.000000000
+ +0100
+++ /home/shmem/perl/lib/Tree/DAG_Node.pm    2018-04-23 03:46:29.60751
+5296 +0200
@@ -1661,7 +1661,7 @@
     if(@daughters) {
       $o->{'_depth'} += 1;
       #print "Depth " , $o->{'_depth'}, "\n";
-      foreach my $one (@daughters) {
+      foreach my $one (sort {$a->name cmp $b->name} @daughters) {
         $one->walk_down($o) if UNIVERSAL::can($one, 'is_node');
         # and if it can do "is_node", it should provide a walk_down!
       }
[download]

result with that:

 root    HASH (0           ): { nc => 0, top_foo => 1 }
    a    HASH (0:0         ): { nc => 1 }
    b    HASH (0:0:0       ): { nc => 2 }
    c    HASH (0:0:2       ): { nc => 6 }
    d    HASH (0:0:2:0     ): { nc => 7 }
    e    HASH (0:0:1       ): { nc => 3 }
    f    HASH (0:0:1:0     ): { nc => 4 }
    g    HASH (0:0:1:0:0   ): { nc => 5 }
    h    HASH (0:1         ): { nc => 8 }
    i    HASH (0:1:0       ): { nc => 9 }
    j    HASH (0:1:0:0     ): { nc => 10 }
[download]

A sort callback for sorting daughters in walk_down would be nice to have. File an enhancement request ;-)

edit: this sorting callback should also be available while populating the 'daughters' anonymous array.

Fun little problem :)

#!/usr/bin/perl

# http://perlmonks.org/?node_id=1213391

use strict;
use warnings;

my @test_data = (
    'a',          #
    '  b',        #
    '  e',        #
    '    f',      #
    '      g',    #
    '  c',        #
    '    d',      #
    'h',          #
    '  i',        #
    '    j',      #
);

my $onestring = join '', map "$_\n", @test_data;
print $onestring, "=" x 60, "\n";

print group($onestring);

sub group
  {
  my @groups;
  push @groups, "$1" . group("$3") while $_[0] =~ /(( *).*\n)((?:\2 +.
+*\n)*)/g;
  join '', sort @groups;
  }
[download]

Why not just

c:\@Work\Perl\monks>perl -wMstrict -le
"my @test_data = (
   'a',
   '  b',
   '  e',
   '    f',
   '      g',
   '  c',
   '    d',
   'h',
   '  i',
   '    j',
   );
 print for @test_data;
 print '===========';
 ;;
 print $_->[0] for
   sort { $a->[1] cmp $b->[1] }
   map [ $_, m{ \w+ }xmsg ],
   @test_data
   ;
"
a
  b
  e
    f
      g
  c
    d
h
  i
    j
===========
a
  b
  c
    d
  e
    f
      g
h
  i
    j
[download]

---

Give a man a fish:  <%-{-{-{-<

Because it fails for

my @test_data = (
    'j',          #
    '  i',        #
    '  h',        #
    '    g',      #
    '      f',    #
    '  e',        #
    '    d',      #
    'c',          #
    '  b',        #
    '    a',      #
);
[download]

G'day atcroft,

Here's a method using non-destructive transliteration. It requires 5.14 (see perl5140delta: Non-destructive substitution). The guts of the solution is:

sort { $a =~ y/ //dr cmp $b =~ y/ //dr } @data
[download]

I've tested with your OP data and tybalt's offering:

#!/usr/bin/env perl

use 5.014;
use warnings;

my @test_data = (
    [ atcroft => [
        'a',          #
        '  b',        #
        '  e',        #
        '    f',      #
        '      g',    #
        '  c',        #
        '    d',      #
        'h',          #
        '  i',        #
        '    j',      #
    ]],
    [ tybalt => [
        'j',          #
        '  i',        #
        '  h',        #
        '    g',      #
        '      f',    #
        '  e',        #
        '    d',      #
        'c',          #
        '  b',        #
        '    a',      #
    ]],
);

say_sorted(@$_) for @test_data;

sub say_sorted {
    my ($who, $data) = @_;
    say "Data from $who";
    say '-' x 20;
    say for @$data;
    say '-' x 20;
    say for sort { $a =~ y/ //dr cmp $b =~ y/ //dr } @$data;
    say '=' x 20;
}
[download]

Output:

Data from atcroft
--------------------
a
  b
  e
    f
      g
  c
    d
h
  i
    j
--------------------
a
  b
  c
    d
  e
    f
      g
h
  i
    j
====================
Data from tybalt
--------------------
j
  i
  h
    g
      f
  e
    d
c
  b
    a
--------------------
    a
  b
c
    d
  e
      f
    g
  h
  i
j
====================
[download]

My understanding is the tree structure should be preserved - only sibling nodes can be swapped to sorted order.
For instance, node 'b' is a child of node 'c' in my original data, and should still be a node of 'c' in the result. It is not.

I think the result should be

c
  b
    a
j
  e
    d
  h
    g
      f
  i
[download]