A note on limits of sequences of binary trees

We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a...

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Published in:Discrete mathematics and theoretical computer science Vol. 25:1; no. Analysis of Algorithms; pp. 1 - 15
Main Author: Grübel, Rudolf
Format: Journal Article
Language:English
Published: Nancy DMTCS 01.01.2023
Discrete Mathematics & Theoretical Computer Science
Subjects:
05c05, 60c05, 68w40
mathematics - combinatorics
mathematics - probability
Metric space
Normality
Permutations
Search algorithms
Set theory
Sorting algorithms
Topology
Trees (mathematics)
ISSN:1365-8050, 1365-8050
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Abstract We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a metric space. For random trees the subtree size topology arises in the context of algorithms for searching and sorting when applied to random input, resulting in a sequence of nested trees. For these we obtain a structural result based on a local version of exchangeability. This in turn leads to a central limit theorem, with possibly mixed asymptotic normality.
AbstractList We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a metric space. For random trees the subtree size topology arises in the context of algorithms for searching and sorting when applied to random input, resulting in a sequence of nested trees. For these we obtain a structural result based on a local version of exchangeability. This in turn leads to a central limit theorem, with possibly mixed asymptotic normality.
We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and permutations we investigate some general aspects of the topology, such as a characterization of the set of possible limits and its structure as a metric space. For random trees the subtree size topology arises in the context of algorithms for searching and sorting when applied to random input, resulting in a sequence of nested trees. For these we obtain a structural result based on a local version of exchangeability. This in turn leads to a central limit theorem, with possibly mixed asymptotic normality.
Author Grübel, Rudolf
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Snippet We discuss a notion of convergence for binary trees that is based on subtree sizes. In analogy to recent developments in the theory of graphs, posets and...
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SubjectTerms 05c05, 60c05, 68w40
mathematics - combinatorics
mathematics - probability
Metric space
Normality
Permutations
Search algorithms
Set theory
Sorting algorithms
Topology
Trees (mathematics)
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