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Bounding right-arm rotation distances
Rotation distance measures the difference in shape between binary trees of
the same size by counting the minimum number of rotations needed to transform
one tree to the other. We describe several types of rotation distance where
restrictions are put on the locations where rotations are permitted, and
provide upper bounds on distances between trees with a fixed number of nodes
with respect to several families of these restrictions. These bounds are sharp
in a certain asymptotic sense and are obtained by relating each restricted
rotation distance to the word length of elements of Thompson's group F with
respect to different generating sets, including both finite and infinite
generating sets.Comment: 30 pages, 11 figures. This revised version corrects some typos and
has some clearer proofs of the results for the lower bounds and better
figure
Bounding right-arm rotation distances
Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets
Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif
A tree is scattered if no subdivision of the complete binary tree is a
subtree. Building on results of Halin, Polat and Sabidussi, we identify four
types of subtrees of a scattered tree and a function of the tree into the
integers at least one of which is preserved by every embedding.
With this result and a result of Tyomkyn, we prove that the tree alternative
property conjecture of Bonato and Tardif holds for scattered trees and a
conjecture of Tyomkin holds for locally finite scattered trees
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