Background: Machine learning methods for clinical assistance require a large number of annotations from trained experts to achieve optimal performance. Previous work in natural language processing has shown that it is possible to automatically extract annotations from the free-text reports associated with chest X-rays. Methods: This study investigated techniques to extract 49 labels in a hierarchical tree structure from chest X-ray reports written in Danish. The labels were extracted from approximately 550,000 reports by performing multi-class, multi-label classification using a method based on pattern-matching rules, a classic approach in the literature for solving this task. The performance of this method was compared to that of open-source large language models that were pre-trained on Danish data and fine-tuned for classification. Results: Methods developed for English were also applicable to Danish and achieved similar performance (a weighted F1 score of 0.778 on 49 findings). A small set of expert annotations was sufficient to achieve competitive results, even with an unbalanced dataset. Conclusions: Natural language processing techniques provide a promising alternative to human expert annotation when annotations of chest X-ray reports are needed. Large language models can outperform traditional pattern-matching methods.