Taxonomy inference for tabular data is a critical task of schema inference, aiming at discovering entity types (i.e., concepts) of the tables and building their hierarchy. It can play an important role in data management, data exploration, ontology learning, and many data-centric applications. Existing schema inference systems focus more on XML, JSON or RDF data, and often rely on lexical formats and structures of the data for calculating similarities, with limited exploitation of the semantics of the text across a table. Motivated by recent works on taxonomy completion and construction using Large Language Models (LLMs), this paper presents two LLM-based methods for taxonomy inference for tables: (i) EmTT which embeds columns by fine-tuning with contrastive learning encoder-alone LLMs like BERT and utilises clustering for hierarchy construction, and (ii) GeTT which generates table entity types and their hierarchy by iterative prompting using a decoder-alone LLM like GPT-4. Extensive evaluation on three real-world datasets with six metrics covering different aspects of the output taxonomies has demonstrated that EmTT and GeTT can both produce taxonomies with strong consistency relative to the Ground Truth.