Looking for example of how to build a custom menu tree

Where can I find example on how to build a custom menu tree? I want to build a module to create a custom menu tree to show on my page header next to the primary links. I want the menu tree to have three items and control each item's visibility on whether the user is login or not.

On the page side, I've figured this much:

<?php theme_menu_tree("my_very_own_menu_tree"); ?>

I would like to know how to code the module side.

Thanks!