In many phylogenetic applications, such as cancer and virus evolution, time trees, evolutionary histories where speciation events are timed, are inferred. Of particular interest are clock-like trees, where all leaves are sampled at the same time and have equal distance to the root. One popular approach to model clock-like trees is coalescent theory, which is used in various tree inference software packages. Methodologically, phylogenetic inference methods require a tree space over which the inference is performed, and the geometry of this space plays an important role in statistical and computational aspects of tree inference algorithms. It has recently been shown that coalescent tree spaces possess a unique geometry, different from that of classical phylogenetic tree spaces. Here we introduce and study a space of discrete coalescent trees. They assume that time is discrete, which is natural in many computational applications. This tree space is a generalisation of the previously studied ranked nearest neighbour interchange space, and is built upon tree-rearrangement operations. We generalise existing results about ranked trees, including an algorithm for computing distances in polynomial time, and in particular provide new results for both the space of discrete coalescent trees and the space of ranked trees. We establish several geometrical properties of these spaces and show how these properties impact various algorithms used in phylogenetic analyses. Our tree space is a discretisation of a previously introduced time tree space, called t-space, and hence our results can be used to approximate solutions to various open problems in t-space.
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