Count comment threads, not total comments in a post

You could try this, must be used in the loop:

<?php

// type = comment will only get "real" comments, no ping-/trackbacks
$comments = get_comments(array('type' => 'comment'));

$threads = 0;
foreach($comments as $comment) {
   // if the comment has no parent, it´s the first of a thread
   if($comment->parent == '') { $threads++; }
}  

switch($threads) {
   case 0: echo "There are no comments so far."; break;
   case 1: echo "There is one thread..."; break;
   default: echo "There are '.$threads.' comment threads going on."; break;
}

?>