Consensus problems for mutation trees

The mutational heterogeneity of tumors can be described with a tree to represent the evolutionary history of the tumor. With noisy sequencing data, there may be uncertainty in the inferred tree structure. The noise inherent in current statistical methods for construction
The evolution of cancer cell mutations presents a significant challenge: dealing with this set of trees to determine a consensus tree that accurately represents the set and to assess the extent of its variability or dispersion. Given a set of mutation trees and the notion of distance, there are at least two natural ways to define the "target" tree, such as a min-sum (Median tree) or a min-max (Closest tree) of a set of trees. Thus, by taking one set of trees as input and dealing with the MEDIAN and CLOSE problems, we prove that both problems are NP-complete, even with only three input trees.