Phylogenetic trees constitute an interesting class of objects for stochastic processes due to the non-standard nature of the space they inhabit. In particular, many statistical applications require the construction of Markov processes on the space of trees, whose cardinality grows superexponentially with the number of leaves considered. We investigate whether certain lower-dimensional projections of tree space preserve the Markov property in tree-valued Markov processes. We study exact lumpability of tree shapes and $\varepsilon$-lumpability of clades, exploiting the combinatorial structure of the SPR graph to obtain bounds on the lumping error under the random walk and Metropolis-Hastings processes. Finally, we show how to use these results in empirical investigation, leveraging exact and $\varepsilon$-lumpability to improve Monte Carlo estimation of tree-related quantities.