Countable 1-transitive trees

We give a survey of three pieces of work, on 2-transitive trees [11], on weakly 2-transitive trees [10], and on lower 1-transitive linear orders [5], all in the countable case. We lead on from these to give a complete description of all the countable 1-transitive trees. In fact the work of [5] was carried out as a preliminary to finding such a description. This is because the maximal chains in any 1-transitive tree are easily seen to be lower 1-transitive, but are not necessarily 1-transitive. In fact a more involved set-up has to be considered, namely a coloured version of the same situation (where ‘colours’ correspond to various types of ramification point), so a major part of what we do here is to describe a large class of countable coloured lower 1-transitive linear orders and go on to use this to complete the description of all countable 1-transitive trees. This final stage involves analyzing how the possible coloured branches can fit together, with particular attention to the possibilities for cones at ramification points.