Private decision tree evaluation is a central component of secure machine learning, as it enables the execution of classification and regression tasks on models while preserving the confidentiality of both user data (features) and model parameters (thresholds). Although recent advances based on homomorphic encryption (HE) provide strong security guarantees, they still suffer from high computational complexity and long inference times, particularly in batch processing scenarios. Moreover, these approaches often require a high degree of interactivity when applied to complex models such as random forests or gradient boosting trees. In this paper, we introduce a new protocol, named StrawHat, which provides an optimized framework for non-interactive batch private evaluation of gradient boosting decision tree models. To the best of our knowledge, this is the first protocol that enables such evaluation in a non-interactive setting. Our approach combines the Row Dichotomy Comparison (RDCMP) comparator with an Oblivious Secure Aggregation (OSA) traversal technique, further leveraging optimized ciphertext aggregation to enhance efficiency. This synergy significantly reduces both computational and communication complexity, while exploiting the benefits of parallel batch processing. Our experiments on real-world datasets demonstrate that StrawHat achieves inference times comparable to server-side evaluation, thereby confirming its fully non-interactive nature: the vast majority of computations are carried out on the server without client intervention. These results pave the way for scalable and efficient privacy-preserving machine learning applications, facilitating the practical deployment of private inference for models such as gradient boosting trees.