Cellular barcoding and other single-cell lineage-tracing strategies form experimental methodologies for analysis of in vivo cell fate that have been instrumental in several significant recent discoveries. Due to the highly nonlinear nature of proliferation and differentiation, interrogation of the resulting data for evaluation of potential lineage pathways requires a new quantitative framework complete with appropriate statistical tests. Here, we develop such a framework, illustrating its utility by analyzing data from barcoded multipotent cells of the blood system. This application demonstrates that the data require additional paths beyond those found in the classical model, which leads us to propose that hematopoietic differentiation follows a loss of potential mechanism and to suggest further experiments to test this deduction. Our quantitative framework can evaluate the compatibility of lineage trees with barcoded data from any proliferating and differentiating cell system.