Visualization of topological constraint rules. (A) A, B, C are three SNVs, each of which represents a set of SNVs with similar SNV population frequencies. When the SNV population frequencies are 0.8, 0.4, 0.2 (left panel), there might be two possible phylogenies that are consistent with these frequencies (middle panel). The two solutions estimate same number of clonal populations but different genotypes for each clone. The decomposition of the clonal population frequencies are shown on the right panel. (B) Because of the sum rule, for this given set of SNV population frequencies 0.8, 0.6, 0.4, a chain structure may be the only possible phylogeny to explain the frequency changes. (C) Under the crossing rule, when multiple samples from the same patient are taken, we would expect the phylogenies are shared between samples. When another set of frequencies are observed 0.8, 0.2, 0.4, the branching structure is the only possible phylogeny to explain the frequencies changes for both sample 1 and 2.