Online discussion threads are conversational cascades in the form of posted messages that can be generally found in social systems that comprise many-to-many interaction such as blogs, news aggregators or bulletin board systems. We propose a framework based on generative models of growing trees to analyse the structure and evolution of discussion threads. We consider the growth of a discussion to be determined by an interplay between popularity, novelty and a trend (or bias) to reply to the thread originator. The relevance of these features is estimated using a full likelihood approach and allows to characterise the habits and communication patterns of a given platform and/or community. We apply the proposed framework on four popular websites: Slashdot, Barrapunto (a Spanish version of Slashdot), Meneame (a Spanish Digg-clone) and the article discussion pages of the English Wikipedia. Our results provide significant insight into understanding how discussion cascades grow and have potential applications in broader contexts such as community management or design of communication platforms.