The advent of single-cell sequencing data provides the ability to model clonal evolution of tumors within individual patients. Inference of such within-patient tumor phylogenies has the potential to advance our understanding of the variation in the process of tumor progression, with strong clinical implications. Here we consider the problem of rapid and robust inference of the tumor phylogeny from a sample of single-cell genotype data under a Markov model that incorporates error in the observed sequencing process. Using techniques from algebraic statistics, we develop a computationally-efficient method for inference and demonstrate its properties using simulation. Further, we describe a method for mapping mutations onto the inferred phylogeny, and show that comparisons of the mapped mutations across patients can lead to insights into the process of cancer evolution. We demonstrate the methods developed using empirical data.