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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
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