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This work concerns the estimation of recursive route choice models in the situation that the trip observations are incomplete, i.e., there are unconnected links (or nodes) in the observations. A direct approach to handle this issue would be intractable because enumerating all paths between unconnected links (or nodes) in a real network is typically not possible. We exploit an expectation-maximization (EM) method that allows to deal with the missing-data issue by alternatively performing two steps of sampling the missing segments in the observations and solving maximum likelihood estimation problems. Moreover, observing that the EM method would be expensive, we propose a new estimation method based on the idea that the choice probabilities of unconnected link observations can be exactly computed by solving systems of linear equations. We further design a new algorithm, called as decomposition-composition (DC), that helps reduce the number of systems of linear equations to be solved and speed up the estimation. We compare our proposed algorithms with some standard baselines using a dataset from a real network and show that the DC algorithm outperforms the other approaches in recovering missing information in the observations. Our methods work with most of the recursive route choice models proposed in the literature, including the recursive logit, nested recursive logit, or discounted recursive models.