Display categories with child category separated by letter

Question

You should simply be able to add another foreach loop, using the get_term_children() function.

Change your lines:

foreach( $tags as $tag ) {
    $name = apply_filters( 'the_title', $tag->name );
    $list .= '<li><a href="' . get_term_link( $tag ) . '" title="' . sprintf(__('View all posts tagged with %s', ''), $tag->name) . '">' . $tag->name . ' </a></li>';
}

…to…

foreach( $tags as $tag ) {
    $name = apply_filters( 'the_title', $tag->name );
    $list .= '<li><a href="' . get_term_link( $tag ) . '" title="' . sprintf(__('View all posts tagged with %s', ''), $tag->name) . '">' . $tag->name . ' </a>';

    // show children in their own ul
    if($children = get_term_children($tag->ID, 'type')) {
        $list .= '<ul>';
        foreach($children as $child) {
            $list .= '<li><a href="' . get_term_link( $child ) . '" title="' . sprintf(__('View all posts tagged with %s', ''), $child->name) . '">' . $tag->name . ' </a></li>';
        }
        $list .= '</ul>';
    }

    // close li after this one
    $list .= '</li>';
}

(untested)

You could make this method far more beautiful if you would do it recursive 😉 But I think this should do.