Computational inference of dated evolutionary histories relies upon various hypotheses about RNA, DNA, and protein sequence mutation rates. Using mutation rates to infer these dated histories is referred to as molecular clock assumption. Coalescent theory is a popular class of evolutionary models that implements the molecular clock hypothesis to facilitate computational inference of dated phylogenies. Cancer and virus evolution are two areas where these methods are particularly important. Methodologically, phylogenetic inference methods require a tree space over which the inference is performed, and geometry of this space plays an important role in statistical and computational aspects of tree inference algorithms. It has recently been shown that molecular clock, and hence coalescent, trees possess a unique geometry, different from that of classical phylogenetic tree spaces which do not model mutation rates. Here we introduce and study a space of discrete coalescent trees, that is, we assume that time is discrete, which is inevitable in many computational formalisations. We establish several geometrical properties of the space and show how these properties impact various algorithms used in phylogenetic analyses. Our tree space is a discretisation of a known time tree space, called t-space, and hence our results can be used to approximate solutions to various open problems in t-space. Our tree space is also a generalisation of another known trees space, called the ranked nearest neighbour interchange space, hence our advances in this paper imply new and generalise existing results about ranked trees.
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