We don't have a supported API for reading the conversation tree for a specific tweet at this time. However, this post from the old google group ( http://groups.google.com/group/twitter-development-talk/browse_thread/thread/cdc34ae78a2350b8 ) details a method called GET statuses/related_results that is an experimental API.
By experimental API, this means that until we officially document it on dev.twitter.com, it's not necessarily production-ready and could be unstable both in the parameters it takes and the format of its responses. It also may just disappear one day.
As for related_results itself, it won't necessarily return every reply for a tweet nor are its responses necessarily limited to just replies. That said, for your own personal use or experimentation you may find some utility in the method. If you choose to use it in any software you're developing, I would proactively wrap its use with significant exception handling.