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Speech recognition systems trained for morphologically rich languages face the problem of vocabulary growth caused by prefixes, suffixes, inflections, and compound words. Solutions proposed in the literature include increasing the size of the vocabulary and segmenting words into morphs. However, in many cases, the methods have only been experimented with low-order n-gram models or compared to word-based models that do not have very large vocabularies. In this paper, we study the importance of using high-order variable-length n-gram models when the language models are trained over morphs instead of whole words. Language models trained on a very large vocabulary are compared with models based on different morph segmentations. Speech recognition experiments are carried out on two highly inflecting and agglutinative languages, Finnish and Estonian. The results suggest that high-order models can be essential in morph-based speech recognition, even when lattices are generated for two-pass recognition. The analysis of recognition errors reveal that the high-order morph language models improve especially the recognition of previously unseen words.