how to create a menu with all sub categories?

Is it possible to create a menu with a category such that each time that I had a sub category that automatically display on the menu?

As far as I know, if I have a menu with the category “fruit”, and I create the subcategory “bananas”, I have to manually add it.

This depends on what kind of menu you are talking about:

1)

If you are talking about “custom menus” (found in the Backend under Design -> Menus) you can do the following:

• Create a new function with the action hook add_category

• inside of this function, you can create a new post of type the menu
  item type, which is added correctly to your menu

• whenever a new category is published the menu item is added now

• you should also hook into delete category, to keep your menu
  up-to-date

2)

You could do much easier without a wordpress menu:

• just write a function to list your categories (or use the predefined
  wp_list_categories function)
• if this is not sufficient write a customized one which (maybe
  recursivley) reads your categories from the database

For option 1 to code you require some knowledge about WordPress though.