  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.
