How to use the right menu_tree and menu_link depending on region How can I have specific menu_tree and menu_link for specific region?

Example:

left region will have his own menu_tree menu_link which will be different from the right region.

How can I achieve this?

Do I need to build a custom module?

If I use template_preprocess_region() , how can I call menu_tree and menu_link?

Thanks in advance

----------- UPDATE ----------------------

I've seen that preprocess_block gives back "region" and "module". Is there anyway to use the different menu_link() and menu_tree() depending on the region and module?
To theme only specific regions within your drupal page simply filter your css by prefixing your class with the class or id of the region in question. You can find the region by inspecting HTML output. I use chromes inspector which is accessed via inspect element within contextual menu.

#region-sidebar-first .menu_tree menu_link_my_class { css }


and if you want a different style for the right hand sidebar 

#region-sidebar-second .menu_tree menu_link_my_class { css }