#!/usr/bin/env perl

use 5.016;
use warnings;

use Archive::Tar;
use Cwd;
use Path::Tiny;

my $src_dir = '/Users/ken/tmp/test_arch/src';
my $zip_dir = '/Users/ken/tmp/test_arch/zip';

my $top_dir = 'fred';
my $zip_name = 'fred.tar.gz';
my $zip_path = path($zip_dir, $zip_name);

{
    use autodie;
    my $cur_dir = getcwd;
    chdir $src_dir or die;
    my @tar_files = ($top_dir);
    path($top_dir)->visit(
        sub { push @tar_files, "$_" }, { recurse => 1 }
    );
    my $tar = Archive::Tar::->new();
    $tar->add_files(@tar_files);
    $tar->write($zip_path, COMPRESS_GZIP);
    chdir $cur_dir or die;
}

##</code><code>##

ken@ganymede: ~/tmp/test_arch/zip
$ ls -al
total 0
drwxr-xr-x 2 ken staff  68 Feb  8 09:55 .
drwxr-xr-x 5 ken staff 170 Feb  8 07:59 ..
ken@ganymede: ~/tmp/test_arch/zip
$ ../bin/arch_fred.pl 
ken@ganymede: ~/tmp/test_arch/zip
$ ls -al
total 4
drwxr-xr-x 3 ken staff 102 Feb  8 09:55 .
drwxr-xr-x 5 ken staff 170 Feb  8 07:59 ..
-rw-r--r-- 1 ken staff 209 Feb  8 09:55 fred.tar.gz
ken@ganymede: ~/tmp/test_arch/zip
$ tar zxvf fred.tar.gz > fred.INSTALL 2>&1
ken@ganymede: ~/tmp/test_arch/zip
$ ls -al
total 8
drwxr-xr-x 5 ken staff 170 Feb  8 09:56 .
drwxr-xr-x 5 ken staff 170 Feb  8 07:59 ..
drwxr-xr-x 6 ken staff 204 Feb  8 07:59 fred
-rw-r--r-- 1 ken staff 104 Feb  8 09:56 fred.INSTALL
-rw-r--r-- 1 ken staff 209 Feb  8 09:55 fred.tar.gz
ken@ganymede: ~/tmp/test_arch/zip
$ cat fred.INSTALL
x fred
x fred/asd
x fred/derf
x fred/qwe
x fred/zxc
x fred/derf/fgh
x fred/derf/rty
x fred/derf/fgh/vbn
ken@ganymede: ~/tmp/test_arch/zip
$ ls -alR fred
fred:
total 0
drwxr-xr-x 6 ken staff 204 Feb  8 07:59 .
drwxr-xr-x 5 ken staff 170 Feb  8 09:56 ..
-rw-r--r-- 1 ken staff   0 Feb  8 07:59 asd
drwxr-xr-x 4 ken staff 136 Feb  8 07:59 derf
-rw-r--r-- 1 ken staff   0 Feb  8 07:59 qwe
-rw-r--r-- 1 ken staff   0 Feb  8 07:59 zxc

fred/derf:
total 0
drwxr-xr-x 4 ken staff 136 Feb  8 07:59 .
drwxr-xr-x 6 ken staff 204 Feb  8 07:59 ..
drwxr-xr-x 3 ken staff 102 Feb  8 07:59 fgh
-rw-r--r-- 1 ken staff   0 Feb  8 07:59 rty

fred/derf/fgh:
total 0
drwxr-xr-x 3 ken staff 102 Feb  8 07:59 .
drwxr-xr-x 4 ken staff 136 Feb  8 07:59 ..
-rw-r--r-- 1 ken staff   0 Feb  8 07:59 vbn
ken@ganymede: ~/tmp/test_arch/zip
$ cd ../..
ken@ganymede: ~/tmp
$ ls -alR fred  # <=== Original files referenced in the OP
fred:
total 0
drwxr-xr-x  6 ken staff  204 Feb  7 11:17 .
drwxr-xr-x 46 ken staff 1564 Feb  8 07:59 ..
-rw-r--r--  1 ken staff    0 Feb  7 11:17 asd
drwxr-xr-x  4 ken staff  136 Feb  7 11:17 derf
-rw-r--r--  1 ken staff    0 Feb  7 11:17 qwe
-rw-r--r--  1 ken staff    0 Feb  7 11:17 zxc

fred/derf:
total 0
drwxr-xr-x 4 ken staff 136 Feb  7 11:17 .
drwxr-xr-x 6 ken staff 204 Feb  7 11:17 ..
drwxr-xr-x 3 ken staff 102 Feb  7 11:18 fgh
-rw-r--r-- 1 ken staff   0 Feb  7 11:17 rty

fred/derf/fgh:
total 0
drwxr-xr-x 3 ken staff 102 Feb  7 11:18 .
drwxr-xr-x 4 ken staff 136 Feb  7 11:17 ..
-rw-r--r-- 1 ken staff   0 Feb  7 11:18 vbn
ken@ganymede: ~/tmp
$