NAME
Graph::Maker::BinaryBeanstalk  create binary beanstalk graph
SYNOPSIS
use Graph::Maker::BinaryBeanstalk;
$graph = Graph::Maker>new ('binary_beanstalk', height => 4);
DESCRIPTION
Graph::Maker::BinaryBeanstalk
creates Graph.pm
graphs of the binary beanstalk per OEIS A179016 etc.
0

1 height => 8 rows
/ \
2 3
/ \
4 5
/ \
6 7
/ \
8 9
/ \
10 11
/ \ / \
12 13 14 15
Vertices are integers starting at root 0. Vertex n has
parent(n) = n  CountOneBits(n)
For example 9 = 1001 binary has 2 1bits so parent 92=7.
Other than the root 0, each vertex has either 0 or 2 children, hence "binary" beanstalk. There are 2 children since if even n has parent nCountOneBits(n)=p then the next vertex n+1 is same
parent(n+1) = n+1  CountOneBits(n+1)
= n+1 = (CountOneBits(n) + 1) since n even
= p
There are no more than 2 children since the next even n+2 has 1bit count
CountOneBits(n+2) <= CountOneBits(n) + 1
equality when n==0 mod 4, otherwise less
due to flipping run of 1bits at second lowest bit position. So parent(n+2) >= n+2  (CountOneBits(n)+1) = p+1, so not the same parent p of n.
This also means parent p is always increasing, and therefore the vertices in a given row are contiguous integers. That's so of the single vertex row 1 and thereafter remains so by parent number increasing.
The vertices in a given row which have children are not always contiguous. The first gap occurs at depth 36 where the vertices 116,117,119 have children and 118 does not.
/^\
112 113
/ \ / \
116 117 118 119 < depth=36
/ \ / \ / \
120 121 122 123 124 125
Options
height
specifies the height of the tree, as number of rows. Height 1 is the root alone, height 2 is two rows being vertices 0 and 1, etc.
N
specifies how many vertices, being vertex numbers 0 to N1 inclusive.
If both height
and N
are given then the tree stops at whichever height
or N
comes first. Since vertex numbers in a row are contiguous, specifying height is equivalent to an N = first vertex number of the row after = 1, 2, 4, 6, 8, ... (OEIS A213708).
FUNCTIONS
$graph = Graph::Maker>new ('binary_beanstalk', key => value, ...)

The key/value parameters are
height => integer N => integer graph_maker => subr(key=>value) constructor, default Graph>new
Other parameters are passed to the constructor, either
graph_maker
orGraph>new()
.Like
Graph::Maker::BalancedTree
, if the graph is directed (the default) then edges are added both up and down between each parent and child. Optionundirected => 1
creates an undirected graph and for it there is a single edge between parent and child.
HOUSE OF GRAPHS
House of Graphs entries for graphs here include
 height=1 (N=1), https://hog.grinvin.org/ViewGraphInfo.action?id=1310 (singleton)
 height=2 (N=2), https://hog.grinvin.org/ViewGraphInfo.action?id=19655 (path2)
 N=3, <https://hog.grinvin.org/ViewGraphInfo.action?id=32234> path3
 height=3 (N=4), https://hog.grinvin.org/ViewGraphInfo.action?id=500 (claw)
 N=5, https://hog.grinvin.org/ViewGraphInfo.action?id=30 (fork)
 height=4 (N=6), https://hog.grinvin.org/ViewGraphInfo.action?id=334 (H graph)
 N=7, https://hog.grinvin.org/ViewGraphInfo.action?id=714
 height=5 (N=8), https://hog.grinvin.org/ViewGraphInfo.action?id=502
 N=13, https://hog.grinvin.org/ViewGraphInfo.action?id=60
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this tree include
http://oeis.org/A179016 (etc)
A011371 parent vertex, nCountOneBits(n)
A213723 child vertex, smaller
A213724 child vertex, bigger
A071542 depth of vertex
A213706 depth of vertex, cumulative
A213708 first vertex in row
A173601 last vertex in row
A086876 row width (run lengths of depth)
A055938 leaf vertices
A005187 nonleaf vertices
A179016 trunk vertices
A213712 trunk increments, = count 1bits of trunk vertex
A213719 trunk vertex predicate 0,1
A213729 trunk vertices mod 2
A213728 trunk vertices mod 2, flip 0<>1
A213732 depths of even trunk vertices
A213733 depths of odd trunk vertices
A213713 nontrunk vertices
A213717 nontrunk nonleaf vertices
A213731 0=leaf, 1=trunk, 2=nontrunk,nonleaf
A213730 start of nontrunk subtree
A213715 trunk position within nonleafs
A213716 nontrunk position within nonleafs
A213727 num vertices in subtree under n (inc self), or 0=trunk
A213726 num leafs in subtree under n (inc self), or 0=trunk
A257126 nth leaf  nth nonleaf
A257130 new high positions of nth leaf  nth nonleaf
A218254 paths to root 0
A213707 positions of root 0 in these paths
A218604 num vertices after trunk in row
A213714 how many nonleaf vertices precede n
A218608 depths where trunk is last in row
A218606 depths+1 where trunk is last in row
A257265 depth down to a leaf, minimum
A213725 depth down to a leaf, maximum in subtree
A218600 depth of n=2^k1
A213709 depth levels from n=2^k1 to n=2^(k+1)1
A213711 how many n=2^k1 blocks preceding given depth
A213722 num nontrunk,nonleaf v between 2^n <= v < 2^(n+1)
SEE ALSO
Graph::Maker, Graph::Maker::BinomialTree
LICENSE
Copyright 2015, 2016, 2017, 2018, 2019 Kevin Ryde
This file is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
This file is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with This file. If not, see http://www.gnu.org/licenses/.