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A general bijective algorithm for trees.

W Y Chen1

  • 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

Proceedings of the National Academy of Sciences of the United States of America
|December 15, 1990
PubMed
Summary
This summary is machine-generated.

This study introduces Schroder trees, a novel combinatorial structure. A key bijection simplifies encoding Schroder trees using integer sequences, with applications in combinatorics and algorithms.

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Area of Science:

  • Combinatorics
  • Graph Theory
  • Computer Science

Background:

  • Trees are fundamental combinatorial structures with applications in decision structures, taxonomy, and computer science.
  • Formal definition of a tree: a connected graph without cycles.
  • Existing tree structures include labeled rooted trees and labeled planar trees.

Purpose of the Study:

  • Introduce Schroder trees, a new class of trees.
  • Establish a bijection between Schroder trees and forests of rooted trees of height one.
  • Demonstrate the utility of this bijection for encoding and algorithmic applications.

Main Methods:

  • Definition of Schroder trees incorporating ordered partitions at vertices.
  • Development of a bijection between Schroder trees and forests of specific rooted trees.
  • Application of the bijection for encoding Schroder trees via integer sequences.

Main Results:

  • A bijection is established between Schroder trees and forests of rooted trees of height one.
  • Schroder trees can be efficiently encoded using sequences of integers.
  • The bijection provides a combinatorial proof for the Lagrange inversion formula.

Conclusions:

  • Schroder trees offer a unified framework for various tree structures.
  • The established bijection simplifies the study and application of Schroder trees.
  • This work has implications for combinatorial algorithms and theoretical computer science.