Binary partition hierarchies and minimum spanning trees are key structures for numerous hierarchical analysis methods, as those involved in computer vision and mathematical morphology. In this article, we consider the problem of their computation in an out-of-core manner, i.e., by minimizing the size of the data structures that are simultaneously needed at the different computation steps. Out-of-core algorithms are necessary when the data are too large to fit entirely in the main memory of the computer, which can be the case with very large images in 2-, 3-, or higher dimension space. We propose a new algebraic framework composed of four main operations on hierarchies: edge-addition, select, insert, and join. Based on this framework, we propose and establish the correctness of an out-of-core calculus for binary partition hierarchies and for minimum spanning trees. First applications to image processing suggest the practical efficiency of this calculus.