23,468 research outputs found
Trees with the most subtrees -- an algorithmic approach
When considering the number of subtrees of trees, the extremal structures
which maximize this number among binary trees and trees with a given maximum
degree lead to some interesting facts that correlate to other graphical indices
in applications. The number of subtrees in the extremal cases constitute
sequences which are of interest to number theorists. The structures which
maximize or minimize the number of subtrees among general trees, binary trees
and trees with a given maximum degree have been identified previously. Most
recently, results of this nature are generalized to trees with a given degree
sequence. In this note, we characterize the trees which maximize the number of
subtrees among trees of a given order and degree sequence. Instead of using
theoretical arguments, we take an algorithmic approach that explicitly
describes the process of achieving an extremal tree from any random tree. The
result also leads to some interesting questions and provides insight on finding
the trees close to extremal and their numbers of subtrees.Comment: 12 pages, 7 figures; Journal of combinatorics, 201
Partitions and Coverings of Trees by Bounded-Degree Subtrees
This paper addresses the following questions for a given tree and integer
: (1) What is the minimum number of degree- subtrees that partition
? (2) What is the minimum number of degree- subtrees that cover
? We answer the first question by providing an explicit formula for the
minimum number of subtrees, and we describe a linear time algorithm that finds
the corresponding partition. For the second question, we present a polynomial
time algorithm that computes a minimum covering. We then establish a tight
bound on the number of subtrees in coverings of trees with given maximum degree
and pathwidth. Our results show that pathwidth is the right parameter to
consider when studying coverings of trees by degree-3 subtrees. We briefly
consider coverings of general graphs by connected subgraphs of bounded degree
Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees
We consider a family of random trees satisfying a Markov branching property.
Roughly, this property says that the subtrees above some given height are
independent with a law that depends only on their total size, the latter being
either the number of leaves or vertices. Such families are parameterized by
sequences of distributions on partitions of the integers that determine how the
size of a tree is distributed in its different subtrees. Under some natural
assumption on these distributions, stipulating that "macroscopic" splitting
events are rare, we show that Markov branching trees admit the so-called
self-similar fragmentation trees as scaling limits in the
Gromov-Hausdorff-Prokhorov topology. The main application of these results is
that the scaling limit of random uniform unordered trees is the Brownian
continuum random tree. This extends a result by Marckert-Miermont and fully
proves a conjecture by Aldous. We also recover, and occasionally extend,
results on scaling limits of consistent Markov branching models and known
convergence results of Galton-Watson trees toward the Brownian and stable
continuum random trees.Comment: Published in at http://dx.doi.org/10.1214/11-AOP686 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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