Tree-based structures can model images—and more generally valued graphs—for processing and analysis purpose. In this framework, the component tree was natively designed for grey-level images—and more generally totally ordered valued graphs. Ten years ago, the notion of a multivalued component tree was introduced to relax this grey-level/total order constraint. In this algorithmic paper, we provide new tools to handle multivalued component trees. Our contributions are twofold: (1) we propose a new algorithm for the construction of the multivalued component tree; (2) we propose two strategies for building hierarchical orders on value sets, required to further build the multivalued component trees of images/graphs relying on such value sets. Codes available at: https://github.com/bnaegel/multivalued_component_tree .