Component-trees are hierarchical structures developed in the framework of mathematical morphology. They model images via the inclusion relationships between the connected components of their successive threshold sets. There exist many variants of component-trees, but to the best of our knowledge, none of them deals with the representation of the image at different scales. In this article, we propose such a variant of component-tree that tackles this issue, namely the Multi-Scale Component-Tree (MSCT). We describe an algorithmic scheme for building the MSCT from the standard computation of component-trees of the image, seen from its lowest to its highest scale. At each step, a local upscaling is performed on relevant parts of the image, corresponding to nodes of the MSCT which are selected according to a stability analysis. The last step builds elements which are part of the standard component-tree (at the highest, native scale of the image). The MSCT provides a compact, efficient representation of images compared to the standard (single-scale) component-tree. In particular, the MSCT is especially suited to analyse images containing sparse objects, which require to be represented at a high scale, vs. large background regions that can be losslessly represented at a lower scale. We illustrate the relevance of the MSCT in the context of cellular image segmentation.