Reconstructing evolutionary histories is crucial across disciplines such as biology and linguistics. Traditionally, research has focused on finding the best phylogenetic tree from large collections, but less attention has been given to treating these collections as distributions within the complex treespace. Advances in software and data have made it possible to infer phylogenies with hundreds or even thousands of taxa, increasing treespace complexity and highlighting the need to view tree collections as samples from broader distributions within treespace. Despite progress, developing statistics over treespace remains challenging due to its complex geometry, limiting the application of conventional methods and leading to reliance on heuristics.

In this thesis, we introduce new statistical methods for analysing ranked and unranked time trees and examine their impact on phylogenetic analyses. We present an algorithm for approximating a mean tree in the space of ranked time trees and explore its properties. Additionally, we extend our results to assess the convergence of phylogenetic Markov chain Monte Carlo (MCMC) analyses by comparing variances of tree distributions. We also revisit the parametrization of posterior tree distributions using conditional clade distributions (CCDs) and show that CCDs accurately estimate the full tree distribution and the mean tree (point estimate). We introduce a new CCD parametrization and highlight that its effectiveness varies with sample size and problem dimensionality. Through extensive simulations and real data applications, we demonstrate that our methods outperform existing state-of-the-art approaches.