Display terms in custom taxonomy child only

I have created custom taxonomies which are working fine. However I am setting up archive pages for each taxonomy. The problem I am having is I would like only the child taxonomy and existing children of the child taxonomy to display on the archive page. I am using the below code. The problem with this code is it displays all custom taxonomies created.

<?php
//list terms in a given taxonomy using wp_list_categories (also useful as a widget if using a PHP Code plugin)

$taxonomy     = 'zip-codes';
$orderby      = 'name';
$show_count   = 0;      // 1 for yes, 0 for no
$pad_counts   = 0;      // 1 for yes, 0 for no
$hierarchical = 1;      // 1 for yes, 0 for no
$title        = '';

$args = array(
  'taxonomy'     => $taxonomy,
  'orderby'      => $orderby,
  'show_count'   => $show_count,
  'pad_counts'   => $pad_counts,
  'hierarchical' => $hierarchical,
  'title_li'     => $title

);
?>

<ul>
<?php wp_list_categories( $args ); ?>
</ul>

How can I show just the children of the child taxonomy of zip-codes.

My current hierarchy is as follows.

Zip Codes
->Colorado Zip Codes
–>80123

Ideally on the page it will only show the numbered zip codes.

Thanks in advanced for all of the help!