wokamoto

You can choose (child) categories and see the posts belong to the chosen category with AJAX.
When the category you choose from the list box has child categories, the child categories will be shown in a new list box. 
Clicking the “Search” button, the list of posts belong to the bottom category which you choose will be displayed under the “Search” button.

Simple Category Search

You can choose (child) categories and see the posts belong to the chosen category with AJAX.

remm

This plugin provide widget that display a list with categories and posts in your sidebar. 
Widget can save the opened categories, and use cache.

Wp Ajax Tree

This plugin provide a widget that display a category and pages tree on your sidebar, using ajax to collapse categories.

Simple Category Search vs Wp Ajax Tree