The coalescent point process of multi-type branching trees

We define a multi-type coalescent point process of a general branching process with countably many types. This multi-type coalescent fully describes the genealogy of the (quasi-stationary) standing population providing types along ancestral lineages of all individuals in the standing population. We show that the coalescent process is a functional of a certain Markov chain defined by the planar embedding of the multi-type branching process.We use the multi-type coalescent process to determine statistical properties of the ancestral tree, such as the time to the most recent common ancestor (MRCA) of two consecutive individuals in the standing population, as well as of two individuals of the same type. These quantities are particularly simple to calculate for branching processes with a multi-type linear-fractional (LF) offspring distribution. We illustrate how an (a)symmetrical offspring distribution affects features of the ancestral tree in an example of a two-type LF branching processes.