Abstract: Organismal development is a complex process that involves repeated cell divisions and gradual differentiation of cells through a hierarchy of progenitor cell types, each with progressively restricted differentiation potential. A central goal in developmental biology is to reconstruct the cell lineage tree – which describes the complete history of cell divisions – and the cell differentiation map – which describes the hierarchy of progenitor cell types and cell type transitions that occur during development. Even for simple organisms such as Caenorhabditis elegans, uncovering the cell lineage tree and the differentiation map of has led to key insights, including the critical role of programmed cell death during development [23]. Creating comparable cell lineage trees and differentiation maps for mammalian development will have far-reaching implications in regenerative medicine and advancement of therapies for developmental disorders. Recent advances in genome editing and single-cell sequencing have enabled high-throughput lineage tracing of cells in complex developmental systems [1, 16, 17, 21]. In these technologies, heritable barcodes are induced in cells, either at specific stages of development [4, 7, 13, 14] or dynamically as cells divide and differentiate [8, 11, 18–20, 22], using genome editing tools such as CRISPR-Cas9. Single-cell RNA sequencing is subsequently used to simultaneously measure barcodes (revealing the lineage of cells) and gene expression (revealing cell types) for thousands of individual cells [2, 5, 6, 9, 10, 12, 15]. While these technologies offer the scalability to investigate development in complex organisms, they do not observe every cell division and the differentiation decisions made by each dividing cell during development. As such, we require computational methods to infer cell lineage trees and differentiation maps from single-cell lineage tracing data. In this talk, we will present two algorithms to infer cell lineage trees and differentiation maps from lineage tracing data. First, we will present Startle [6], an algorithm to infer cell lineage trees from CRISPR-Cas9-based lineage tracing. Startle introduces the star homoplasy evolutionary model that constrains a phylogenetic character to mutate at most once along a lineage, capturing the non-modifiability property of CRISPR-Cas9 mutations. We derive a combinatorial characterization of star homoplasy phylogenies and use this characterization in Startle to compute a maximum parsimony star homoplasy phylogeny from lineage tracing data. The second algorithm, CARTA [3], infers cell differentiation maps from lineage tracing data in which observed and unobserved progenitor cell types are represented by their potency. CARTA introduces a novel mathematical framework that balances the trade-off between the complexity of the differentiation map and the fit of the map with the cell lineage trees derived from lineage tracing data. We also introduce a formal quantitative framework to systematically assess different models of cell differentiation maps with varying numbers of progenitors. Throughout the talk, we will demonstrate that Startle and CARTA outperform existing methods on both simulated and real data. Overall, my talk will underscore the crucial role of specialized models and algorithms to effectively analyze lineage tracing data and uncover the dynamics of developmental biological systems.