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The classification service over a stream of data is becoming an important offering for cloud providers, but users may encounter obstacles in providing sensitive data due to privacy concerns. While Trusted Execution Environments (TEEs) are promising solutions for protecting private data, they remain vulnerable to side-channel attacks induced by data-dependent access patterns. We propose a Privacy-preserving Data Stream Training and Inference scheme, called EnclaveTree, that provides confidentiality for user's data and the target models against a compromised cloud service provider. We design a matrix-based training and inference procedure to train the Hoeffding Tree (HT) model and perform inference with the trained model inside the trusted area of TEEs, which provably prevent the exploitation of access-pattern-based attacks. The performance evaluation shows that EnclaveTree is practical for processing the data streams with small or medium number of features. When there are less than 63 binary features, EnclaveTree is up to ${\thicksim}10{\times}$ and ${\thicksim}9{\times}$ faster than naïve oblivious solution on training and inference, respectively.