display childs without parent

The following code displays the parent plus his 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 childre/ 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] = "<pre><a href=\"{$ url}\">{$ category->name}</a></pre>"; } echo "" . implode("\n",$ categories) . ""; ?>