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Today, machine learning is widely applied in sensitive, security-related, and financially lucrative applications. Model extraction attacks undermine current business models where a model owner sells model access, e.g., via MLaaS APIs. Additionally, stolen models can enable powerful white-box attacks, facilitating privacy attacks on sensitive training data, and model evasion. In this paper, we focus on Decision Trees (DT), which are widely deployed in practice. Existing black-box extraction attacks for DTs are either query-intensive, make strong assumptions about the DT structure, or rely on rich API information. To limit attacks to the black-box setting, CPU vendors introduced Trusted Execution Environments (TEE) that use hardware-mechanisms to isolate workloads from external parties, e.g., MLaaS providers. We introduce TrEEStealer, a high-fidelity extraction attack for stealing TEE-protected DTs. TrEEStealer exploits TEE-specific side-channels to steal DTs efficiently and without strong assumptions about the API output or DT structure. The extraction efficacy stems from a novel algorithm that maximizes the information derived from each query by coupling Control-Flow Information (CFI) with passive information tracking. We use two primitives to acquire CFI: for AMD SEV, we follow previous work using the SEV-Step framework and performance counters. For Intel SGX, we reproduce prior findings on current Xeon 6 CPUs and construct a new primitive to efficiently extract the branch history of inference runs through the Branch-History-Register. We found corresponding vulnerabilities in three popular libraries: OpenCV, mlpack, and emlearn. We show that TrEEStealer achieves superior efficiency and extraction fidelity compared to prior attacks. Our work establishes a new state-of-the-art for DT extraction and confirms that TEEs fail to protect against control-flow leakage.