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A Maximum Agreement SubTree (MAST) is a largest subtree common to a set of trees and serves as a summary of common substructure in the trees. A single MAST can be misleading, however, since there can be an exponential number of MASTs, and two MASTs for the same tree set do not even necessarily share any leaves. In this paper, we introduce the notion of the Kernel Agreement SubTree (KAST), which is the summary of the common substructure in all MASTs, and show that it can be calculated in polynomial time (for trees with bounded degree). Suppose the input trees represent competing hypotheses for a particular phylogeny. We explore the utility of the KAST as a method to discern the common structure of confidence, and as a measure of how confident we are in a given tree set. We also show the trend of the KAST, as compared to other consensus methods, on the set of all trees visited during a Bayesian analysis of flatworm genomes.