Mod Articles Count
A module extension for the Joomla 1.5, 2.5 and 3 and 4 CMS. The articles count module (as the name suggests), counts the articles in your Joomla categories, and displays them in a list with a link to the category. You can choose which categories to display, or use the currently displayed category. There is a choice of display: either as a simple list or a collapsible tree.
Since version 3.0.0 the module supports the Joomla core extensions com_contacts, com_newsfeeds and com_weblinks, as well as the core com_content component. Also there is a version available which counts K2 articles in categories in exactly the same way.
You can find out more about the module here.
Optionally the module can use a collapsible tree to show the category structure, as shown below:-
Articles Count
This is a free module released under the GPL, the download links are below.