Abstract
LetR n denote the set of rooted trees withn leaves in which: the leaves are labeled by the integers in {1, ...,n}; and among interior vertices only the root may have degree two. Associated with each interior vertexv in such a tree is the subset, orcluster, of leaf labels in the subtree rooted atv. Cluster {1, ...,n} is calledtrivial. Clusters are used in quantitative measures of similarity, dissimilarity and consensus among trees. For anyk trees inR n , thestrict consensus tree C(T 1, ...,T k ) is that tree inR n containing exactly those clusters common to every one of thek trees. Similarity between treesT 1 andT 2 inR n is measured by the numberS(T 1,T 2) of nontrivial clusters in bothT 1 andT 2; dissimilarity, by the numberD(T 1,T 2) of clusters inT 1 orT 2 but not in both. Algorithms are known to computeC(T 1, ...,T k ) inO(kn 2) time, andS(T 1,T 2) andD(T 1,T 2) inO(n 2) time. I propose a special representation of the clusters of any treeT R n , one that permits testing in constant time whether a given cluster exists inT. I describe algorithms that exploit this representation to computeC(T 1, ...,T k ) inO(kn) time, andS(T 1,T 2) andD(T 1,T 2) inO(n) time. These algorithms are optimal in a technical sense. They enable well-known indices of consensus between two trees to be computed inO(n) time. All these results apply as well to comparable problems involving unrooted trees with labeled leaves.
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The Natural Sciences and Engineering Research Council of Canada partially supported this work with grant A-4142.
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Day, W.H.E. Optimal algorithms for comparing trees with labeled leaves. Journal of Classification 2, 7–28 (1985). https://doi.org/10.1007/BF01908061
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DOI: https://doi.org/10.1007/BF01908061