Internet boards are platforms for online discussion about a variety of topics. On these boards, individuals may start a new thread on a specific matter or leave comments in an existing discussion. The resulting collective process leads to the formation of discussion trees', where nodes represent a post and comments, and an edge represents a reply-to' relation. The structure of discussion trees has been analysed in previous works, but mainly from a static perspective. In this article, we focus on the structural and dynamical properties of discussion trees by modelling their formation as a self-exciting Hawkes process with a branching structure. We first study a Reddit data set of all posts and comments submitted between 2008 and 2014 to show that the structure of the trees resemble those produced by a Galton-Watson process with a special root offspring distribution. The dynamical aspect of the model is then used to predict future commenting activity and the final size of a discussion tree. We compare the efficiency of our approach with previous works and show its superiority for the prediction of discussion dynamics. In particular, we find that the Reddit discussion trees, being fundamentally broader than for example the Twitter cascades and developing with a qualitatively different dynamics, require appropriate models. The tree structure and the dynamics are influenced mainly by the age of the discussion, rather than by the number of acquired comments. The post and the comments are further commented by different processes, which is a key ingredient in modelling of discussions.