Add links to an existing custom menu

I want to add a tree of links to a custom menu that already exists (created in admin/build/menu). I want to do this programatically based on an array I have. I don't maintain the menu but if it's there I need certain links to exist in it.

Given:

• Menu A
  • Item A
  • Item B

I want:

• Menu A
  • Item A
  • Item B
  • Item C
    • Item C.1
    • Item C.2
      • Item C.2.a
      • Item C.2.b
      • Item C.2.c
      • Item C.2.d
    • Item C.3

What's the best way to go about this? I've tried hook_menu and hook_menu_alter and can get my items to show in the menu but not according to their tree design. My $items array looks like this:

<?php

$items['C'] = array(...);
$items['C/1'] = array(...);
$items['C/2'] = array(...);
$items['C/2/a'] = array(...);
$items['C/2/b'] = array(...);
$items['C/2/c'] = array(...);
$items['C/2/d'] = array(...);
$items['C/3'] = array(...);

?>

All items are showing up on the same level though. What should I be specifying in array(...) that actually makes the tree? I was under the impression that the menu module based the tree structure on the keys of the $items array.

Thanks!