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Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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The inverse problem for certain tree parameters.

Eva Czabarka1, László Székely, Stephan Wagner

  • 1Department of Mathematics, University of South Carolina, Columbia, SC 29208, United States of America.

Discrete Applied Mathematics (Amsterdam, Netherlands : 1988)
|February 18, 2010
PubMed
Summary
This summary is machine-generated.

Researchers explored the inverse problem for graph parameters, specifically focusing on the number of subtrees in trees. They proved that nearly all positive integers can be realized as the number of subtrees for some tree.

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Area of Science:

  • Graph theory
  • Discrete mathematics
  • Combinatorics

Background:

  • Graph parameters assign integer values to graphs.
  • The inverse problem seeks a graph given a parameter value.
  • Focus is on trees and the number of subtrees parameter.

Purpose of the Study:

  • Investigate the existence of trees for given subtree counts.
  • Establish a general framework for inverse problems on trees.
  • Determine which integers are representable as the number of subtrees.

Main Methods:

  • General theoretical framework for inverse graph problems on trees.
  • Analysis of the 'number of subtrees' parameter.
  • Reduction to number-theoretic considerations.

Main Results:

  • Every positive integer, except for 34 specific exceptions, is the number of subtrees of some tree.
  • Demonstrated a general approach for inverse problems on trees.
  • Identified the exceptions to the general rule for subtree counts.

Conclusions:

  • The number of subtrees parameter is widely surjective over the set of trees.
  • The inverse problem for the number of subtrees is largely solvable.
  • Number theory plays a crucial role in solving inverse graph parameter problems.