I have also downloaded all_sub_cats_products, so I now have all_sub_cat_products and simple_category_tree.

So, I'm guessing that I need the latter in place before the former? ...and that the following files are required:

includes/modules/sideboxes/my_categories.php
includes/templates/template_default/sideboxes/tpl_my_categories.php

...and that my_categories.php needs to contain

Code:
includes/modules/sideboxes/my_categories.php
PHP Code:
<?php
    require($template->get_template_dir('tpl_categories.php',DIR_WS_TEMPLATE, $current_page_base,'sideboxes'). '/tpl_my_categories.php');

    $title = 'Simple Categories Tree Demo';
    $title_link = false;

    require($template->get_template_dir($column_box_default, DIR_WS_TEMPLATE, $current_page_base,'common') . '/' . $column_box_default);

?>

includes/templates/template_default/sideboxes/tpl_my_categories.php
PHP Code:
<?php
$content = '';
$content .= $_SESSION['category_tree']->build_category_string('ul', 'li', '', 0, 2);
?>
...whilst includes/templates/template_default/sideboxes/tpl_my_categories.php remains blank?

I am not sure what use the all_sub_cats_products would then be.