Add classes wp_list_categories

functions.php

//themes functions.php

class MyWalker extends Walker_Category {
 
	function start_el(&$output, $category, $depth, $args) {
		extract($args);

		$cat_name = esc_attr( $category->name );
		
		$link = '<a href="%1$s" rel="%2$s">%3$s</a>';
		
		$link = sprintf($link
							, esc_attr( get_term_link($category) )
							, $category->slug
							, $cat_name
							);
		$check_mago = count( get_ancestors( $category->term_id, 'category' ) );
			
		if($check_mago == 2 and $category->category_parent > 0 ){
			$class = 'mago';
		}
		if($check_mago == 1 and $category->category_parent > 0 ){
			$class = 'kodomo';
		}
		if($category->category_parent == 0){
			$class = 'oya';
		}	
		$output .= '<li class="'.$class.'">'.$link;
	}
}

//template

<?php	
$MyWalker = new MyWalker();
$args = array('walker' => $MyWalker);
?>
<ul>
<?php
wp_list_categories($args);
?>
</ul>