Elucidating the dynamics of cellular differentiation in space and time is key to advancing organoid biology and technology. Direct visualization of differentiation patterns is challenging, however, in part because of the difficulty of simultaneously detecting all relevant cell types by fluorescence imaging. Here we present TypeTracker, which determines the differentiation trajectories of all cells within a region of interest, including their type transitions, growth, divisions, and changes in spatial organization. We show how the lineage tree topology, as determined by automated cell tracking, allows for the retrospective identification of cell types across multiple generations. The data revealed various surprising aspects of intestinal organoid differentiation, including symmetric fate adoption by sister cells, cell type abundance regulated by division, and type commitment of cells occurring prior to their spatial reorganization. Our method can be applied broadly to study other organoid systems and to screen for compounds that affect differentiation programs.