display childs without parent

The following code displays the parent plus its children on the specific parent - and children-category pages as filter-buttons. But in this list I do not want to display the parent, (because the parent is already displayed as ALL). How can I display list with only the children without the parent?

Thank you for any further help.

?php
    $categories = get_the_category();
    foreach($categories as $key = $category) {
        $url = get_term_link((int)$category-term_id,'category');
        $categories[$key] = prea href=\{$url}\{$category-name}/a/pre;
    }
    echo  . implode(\n,$categories) . ;
?