List custom taxonomy with children

I’m trying to create a list of links for a custom taxonomy along with the children of the terms. i.e:

<ul>
	<li><a href="/telephony">Telephony</a>
		<ul>
	    	<li><a href="/blackberry">BlackBerry</a></li>
	    	<li><a href="/fixed-ip">Fixed IP</a></li>
		</ul>
	</li>
   	<li><a href="/email">Email</a>
		<ul>
	    	<li><a href="/fax">Fax</a></li>
	    	<li><a href="/text">Text</a></li>
	    	<li><a href="/nhs-email">NHS Email</a></li>
		</ul>
	</li>
</ul>

So far I’ve got:

<?php
	$termID = 451;
	$taxonomyName = "service_line_category";
	$termchildren = get_term_children( $termID, $taxonomyName );

	echo '<ul>';
	foreach ($termchildren as $child) {
		$term = get_term_by( 'id', $child, $taxonomyName );
		echo '<li><a href="' . get_term_link( $term->name, $taxonomyName ) . '">' . $term->name . '</a></li>';
	}
	echo '</ul>';
?>

However this only grabs the children of ‘$termID = 451;’ but I’m not sure how to modify this to get the parents and the children for all of the terms within ‘service_line_category’ in a list.

Any ideas? I’m not really sure where to go with this.