I would use a recursive subroutine to walk the tree and collect the output as you go. Maybe something like this will help you (depth traversal):

use HTML::Template;

my $thingy = [
    {
        name     => 'foo',
        id       => 1,
        children => [
            {
                name     => 'bar',
                id       => 2,
                children => [
                    {
                        name => 'baz',
                        id   => 3,
                    },
                    {
                        name => 'qux',
                        id   => 4,
                    },
                ],
            },
            {
                name => 'ducks',
                id   => 5,
            },
        ],
    },
    {
        name     => 'last',
        id       => 6,
    },
];

our $tmpl = HTML::Template->new(
    filehandle        => \*DATA,
    die_on_bad_params => 0,
);
our $output;
our $tab_loop = [];

recurse($thingy);
print $output;

sub recurse {
    my $thingy = shift;

    for (@$thingy) {
        $tmpl->param(tab_loop => $tab_loop, %$_);
        $output .= $tmpl->output;
        push @$tab_loop, { tab => "\t" };
        recurse($_->{children}) if ref $_->{children};
        pop @$tab_loop;
    }
}

__DATA__
<tmpl_loop tab_loop><tmpl_var tab></tmpl_loop><tmpl_var id> - <tmpl_va
+r name>
[download]

I'd be tempted to attack this problem with Template instead.

L-LL-L--L-LL-L--L-LL-L--
-R--R-RR-R--R-RR-R--R-RR
B--B--B--B--B--B--B--B--
H---H---H---H---H---H---
(the triplet paradiddle with high-hat)