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Extremal distances for subtree transfer operations in binary trees. (English) Zbl 1414.05074
Summary: Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection (TBR), subtree prune and regraft (SPR), and rooted subtree prune and regraft (rSPR). We show that for a pair of leaf-labelled binary trees with $$n$$ leaves, the maximum number of such moves required to transform one into the other is $$n-\Theta (\sqrt{n})$$, extending a result of Y. Ding et al. [J. Comb. Theory, Ser. A 118, No. 7, 2059–2065 (2011; Zbl 1231.05072)], and this holds also if one of the trees is fixed arbitrarily. If the pair is chosen uniformly at random, then the expected number of moves required is $$n-\Theta (n^{2/3})$$. These results may be phrased in terms of agreement forests: we also give extensions for more than two binary trees.

##### MSC:
 05C05 Trees 05C12 Distance in graphs 05C35 Extremal problems in graph theory 05C76 Graph operations (line graphs, products, etc.) 92D15 Problems related to evolution
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