# Optimal algorithms for comparing trees with labeled leaves

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## Abstract

Let*R*_{ n } denote the set of rooted trees with*n* 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 vertex*v* in such a tree is the subset, or*cluster*, of leaf labels in the subtree rooted at*v.* Cluster {1, ...,*n*} is called*trivial*. Clusters are used in quantitative measures of similarity, dissimilarity and consensus among trees. For any*k* trees in*R*_{ n }, the*strict consensus tree C*(*T*_{1}, ...,*T*_{ k }) is that tree in*R*_{ n } containing exactly those clusters common to every one of the*k* trees. Similarity between trees*T*_{1} and*T*_{2} in*R*_{ n } is measured by the number*S*(*T*_{1},*T*_{2}) of nontrivial clusters in both*T*_{1} and*T*_{2}; dissimilarity, by the number*D*(*T*_{1},*T*_{2}) of clusters in*T*_{1} or*T*_{2} but not in both. Algorithms are known to compute*C*(*T*_{1}, ...,*T*_{ k }) in*O*(*kn*^{2}) time, and*S*(*T*_{1},*T*_{2}) and*D*(*T*_{1},*T*_{2}) in*O*(*n*^{2}) time. I propose a special representation of the clusters of any tree*T R*_{ n }, one that permits testing in constant time whether a given cluster exists in*T*. I describe algorithms that exploit this representation to compute*C*(*T*_{1}, ...,*T*_{ k }) in*O*(*kn*) time, and*S*(*T*_{1},*T*_{2}) and*D*(*T*_{1},*T*_{2}) in*O*(_{n}) time. These algorithms are optimal in a technical sense. They enable well-known indices of consensus between two trees to be computed in*O*(*n*) time. All these results apply as well to comparable problems involving unrooted trees with labeled leaves.

## Keywords

Algorithm complexity Algorithm design Comparing hierarchical classifications Comparing phylogenetic trees Consensus index Strict consensus tree## Preview

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