The task of assigning diagnostic ICD codes to patient hospital admissions is typically performed by expert human coders. Efforts towards automated ICD coding are dominated by supervised deep learning models. However, difficulties in learning to predict the large number of rare codes remain a barrier to adoption in clinical practice. In this work, we leverage off-the-shelf pre-trained generative large language models (LLMs) to develop a practical solution that is suitable for zero-shot and few-shot code assignment. Unsupervised pre-training alone does not guarantee precise knowledge of the ICD ontology and specialist clinical coding task, therefore we frame the task as information extraction, providing a descriptionof each coded concept and asking the model to retrieve related mentions. For efficiency, rather than iterating over all codes, we leverage the hierarchical nature of the ICD ontology to sparsely search for relevant codes. Then, in a second stage, which we term ‘meta-refinement’, we utilise GPT-4 to select a subset of the relevant labels as predictions. We validate our method using Llama-2, GPT-3.5 and GPT-4 on the CodiEsp dataset of ICD-coded clinical case documents. Our tree-search method achieves state-of-the-art performance on rarer classes, achieving the best macro-F1 of 0.225, whilst achieving slightly lower micro-F1 of 0.157, comparedto 0.216 and 0.219 respectively from PLM-ICD. To the best of our knowledge, this is the first method for automated ICD coding requiring no task-specific learning.