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Concept bottleneck models (CBM) aim to improve model interpretability by predicting human level “concepts” in a bottleneck within a deep learning model architecture. However, how the predicted concepts are used in predicting the target still either remains black-box or is simplified to maintain interpretability at the cost of prediction performance. We propose to use Fast Interpretable Greedy Sum- Trees (FIGS) to obtain Binary Distillation (BD). This new method, called FIGSBD, distills a binary-augmented concept-to-target portion of the CBM into an interpretable tree-based model, while maintaining the competitive prediction performance of the CBM teacher. FIGS-BD can be used in downstream tasks to explain and decompose CBM predictions into interpretable binary-concept-interaction attributions and guide adaptive test-time intervention. Across 4 datasets, we demonstrate that our adaptive test-time intervention identifies key concepts that significantly improve performance for realistic human-in-the-loop settings that only allow for limited concept interventions. All code is made available on Github (https://github.com/mattyshen/adaptiveTTI).