Relations between adjacency trees

Adjacency trees can model the nesting structure of spatial regions. In many applications it is necessary to model foreground and background regions which exhibit changes over time such as splitting, where one region divides into two. For example, the qualitative description of the development of wildfires would use the foreground for areas on fire and the background for areas not on fire. Such dynamic behaviour can be modelled by a particular kind of relation between the nodes of two adjacency trees representing the initial and final configurations of the regions at two times. These relations, which we call bipartite, correspond to having an arbitrary relation between the foreground regions at the two times and an arbitrary relation between the background regions at the two times. We show that all bipartite relations between trees arise from sequences of atomic relations between trees. There are just four types of these atomic relations (in addition to one representing no change): inserts, splits, merges and deletes.