Well, now I have a sight change in my problem.

The array @path needs to keep a record of all the keys it went through to get to level it is on the tree. Below I got this kind of working. The array @path needs to be poped again if the sub is on the last item in a group. I can't explain this very well at all so you might want to try the sub out yourself (with the above hash) :)

{
    my @path;
    sub hash_gen {
        my $cats = shift;
        unless (ref $cats eq 'HASH') {
            pop @path;
            return undef;
        }
        my ($output, $level) = ('', 0);
        foreach my $cat (sort {uc($a) cmp uc($b)} keys % $cats) {
            push @path, $cat;
            $output .= '<li>' . join(' -> ', @path) . "</li>\n";
            pop @path;
            $output .= hash_gen($cats->{$cat}) . "\n";
        }
        return "<ul>$output</ul>";
    }
}
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