Abstract

Branching processes or Galton-Watson processes can be used to model the genealogy of a population of different species, where birth and death events represent speciation and extinction. In the more general context of multi-type branching processes, species are classified under phenotypical traits, and the probability of speciation and extinction is dependent on individual types. Since most accessible biological data concerns surviving species, it becomes necessary to extract information about the shape of genealogical trees from the available knowledge on the standing population, and to devise random models allowing backward reconstruction of ancestry under the rules of a particular branching process. We present two investigations on the topology of ancestral multi-type branching trees, generalizing several known results from the single-type case, and obtaining some new results that can only be formulated in the multi-type setting.

In the first part of the thesis, we present a backward construction algorithm for the ancestral tree of a planar embedding of a multi-type Galton-Watson tree assumed to be quasi-stationary, and we derive formulae for the conditional distribution of the time to the most recent common ancestor of two consecutive individuals at the present time, and of two individuals of the same type.
We specialize these formulae to multi-type linear-fractional branching processes, and observe some effects of the symmetry of the parameters in the two-type case.

In the second part of the thesis, we extend the concepts of cherries and pendant edges from rooted binary trees to the multi-type setting, and compute expressions and asymptotic properties for mean numbers and variances of these structures under the neutral two-type Yule model.

We explain how type mutations appear naturally in ancestral trees of multi-type birth-death processes, and show that these ancestral trees are Markovian and behave as pure-birth processes, by giving explicit time-dependent rates. We derive formulae and asymptotic properties for the mean number of cherries and pendant edges of each type in a multi-type pure-birth process with mutations. We show that sometimes it is possible to recover the defining rates of this process from the asymptotic proportion of leaves, cherries and pendant edges of each type.