Example code for overriding specific menu in Drupal 7 using theme_menu_tree()

template.php

<?php

/**
 * Implements hook_menu_tree
 * Wrap main menu with 2 division (Bootstrap)
 *
 */
function speedy_menu_tree__main_menu($vars) {
  return '<div class="navbar"><div class="navbar-inner"><ul class="nav">' . $vars['tree'] . '</ul></div></div>';
}

/**
 * Implements hook_menu_tree
 * Remove class for user menu
 *
 */
function speedy_menu_tree__user_menu($vars) {
  return '<ul>' . $vars['tree'] . '</ul>';
}

Please note, the word speedy in my function name is the name of my theme. Make sure to replace that with the correct name, this examples uses theme_menu_tree() function.