Black-box verifiers for abstractive summaries often struggle with complex claims that require multi-hop reasoning, and they typically provide a single verdict without an interpretable rationale. As a result, it becomes difficult to understand or audit their failures. We address this with HalluTree, a framework that models verification as an interpretable claim tree. HalluTree first decomposes summaries into subclaims, classifying each into two types – extractive (directly verifiable against evidence) or inferential (requiring reasoning) – which follow distinct verification paths. Extractive claims are robustly verified against evidence using an ensemble of lightweight NLI models. Crucially, inferential claims trigger a process that generates a natural program – an explicit reasoning chain that integrates supporting evidence and logical steps – which is then executed to determine the claim’s validity. Evaluation on the LLM-AggreFact benchmark demonstrates HalluTree’s effectiveness: it achieves performance competitive with top-tier black-box models, including Bespoke-MiniCheck, while providing transparent and auditable reasoning programs for every inferential judgment. This combination of competitive accuracy and high interpretability offers a significant advance over opaque, single-classification verifiers. We will publically release code, data, prompts, and other artifacts upon acceptance.