Traverses and return all the nested children post names of a root page.

$children contains parent-children relations

function _page_traverse_name( $page_id, &$children, &$result ) {
	if ( isset( $children[ $page_id ] ) ) {
		foreach ( (array) $children[ $page_id ] as $child ) {
			$result[ $child->ID ] = $child->post_name;
			_page_traverse_name( $child->ID, $children, $result );
		}
	}
}