Two dominant tokenisation algorithms are used by modern language models: byte-pair encoding (BPE) and UnigramLM. These differ along two orthogonal axes: their optimisation objective (compression vs. log-likelihood) and their search procedure (bottom-up merging vs. top-down pruning). Existing comparisons confound these axes, making it unclear whether their observed differences stem from what is being optimised vs. how it is being optimised. We disentangle the two by introducing two new tokenisation algorithms that complete this 2x2 design space: BottomUpLL, a bottom-up likelihood-based tokeniser, and TopDownComp, a top-down compression-based tokeniser. We train language models with tokenisers produced by each algorithm, varying: model size, vocabulary sizes, and domain (English-only vs. multilingual). Evaluating models on bits-per-byte, we find that the search procedure -- not the objective -- is the dominant factor: bottom-up tokenisers consistently achieve lower bits-per-byte in most settings. Evaluating models on the BLiMP task, however, shows no consistent relationship between design choice and performance. Overall, our results disentangle the effect of tokeniser design choices on language modelling performance, offering concrete guidance for their more principled construction.
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