Retrieve post of child category

Hello,

I want to retrieve post of a child category, by it’s parent category?
What I mean?
Let’s say I have a subcategory name dogs, and it’s parent name animals.
What I want is to retrieve the posts of dogs category, and it’s need to be parent category-depended.
I saw this code on the web, it help a bit, but it retrieve all posts of all childs, and I want only of a specific child category.
This is the code:

<?php
//get all posts for children of category $cata
$cata = $cat_id;
$taxonomy = 'category';
$cata_children = get_term_children( $cata, $taxonomy );
$args=array(
 'category__in' => $cata_children,
);
$my_query = null;
$my_query = new WP_Query($args);
if( $my_query->have_posts() ) {
echo 'List of Posts belonging to Category A children';
while ($my_query->have_posts()) : $my_query->the_post(); ?>
 <p><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link  to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></p>
<?php
endwhile;
}
wp_reset_query();  // Restore global post data stomped by the_post().
?>

Any help? Thanks!