Count posts of child category in parent category

Hallo WordPress Community,

I have a Woocommerce shop site with multiple authors. In the sidebar I want to display a list of woocommerce product categories by author (based on posts which an author made), accompanied by a product post counter. I use following code by 2famousTV, which works fine to get the categories by author and a counter. I modified it so it only shows parent categories.

<?php

global $post, $wpdb;

// This will get us a list of the categories that our Author has published in
$author = get_query_var('author');
$categories = $wpdb->get_results("

SELECT DISTINCT(terms.term_id) as ID, terms.name, terms.slug, tax.description
FROM $wpdb->posts as posts
LEFT JOIN $wpdb->term_relationships as relationships ON posts.ID = relationships.object_ID
LEFT JOIN $wpdb->term_taxonomy as tax ON relationships.term_taxonomy_id = tax.term_taxonomy_id
LEFT JOIN $wpdb->terms as terms ON tax.term_id = terms.term_id
WHERE posts.post_status = 'publish' AND
posts.post_author = '$author' AND
tax.taxonomy = 'product_cat'
ORDER BY terms.name ASC
");

// This loop picks up categories
foreach($categories as $category) : 

$catid = $category->ID;

// Now, inside the loop, we need to count how many posts that the Author has published.
$counter = "SELECT COUNT(*)
FROM $wpdb->posts
LEFT JOIN $wpdb->term_relationships ON($wpdb->posts.ID =     $wpdb->term_relationships.object_id)
LEFT JOIN $wpdb->term_taxonomy ON($wpdb->term_relationships.term_taxonomy_id = $wpdb->term_taxonomy.term_taxonomy_id)
WHERE $wpdb->term_taxonomy.term_id = $catid
AND $wpdb->term_taxonomy.taxonomy = 'product_cat'

// modification  

AND tax.parent = '0'

// modification end

AND $wpdb->posts.post_status = 'publish'
AND post_author = '$author'
";

$user_count = $wpdb->get_var($counter);

echo '<div class="archive_author">' . $category->name . '<br/><span    class="subcounter">' . $user_count . ' posts</span></div>';

endforeach; 

?>

Source: Link to original post by 2famous-tv

My question:

With this code the parent category counter only counts items within the parent category but not the items within its child categories. How can I make the parent category counter include the posts of the child category without actually assigning the posts to the parent category?

Category________________Counter___________Posts

Parent-Category___________0__________________0

-Child-Category A_________2__________________2

-Child-Category B_________3__________________3

should be:

Category_______________Counter___________Posts

Parent-Category___________5__________________0

-Child-Category A_________2__________________2

-Child-Category B_________3__________________3

I tried it with several WordPress inherid options like “wp_list_categories”, “get_terms” and “get_the_term” but did not bring results. It seems the only way to go through Mysql. I am not firm enough to make that happen. Any help would be appriciated!