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 15Vertices are integers starting at root 0. Vertex n has
parent(n) = n - CountOneBits(n)
= 0,0,1,1,3,3,4,4,7,7,8,8,,... (A011371)For example 9 = 1001 binary has 2 1-bits so parent 9-2=7.
Other than the root 0, each vertex has 0 or 2 children, hence "binary" beanstalk. There are 2 children (not 1) since if even n has parent n-CountOneBits(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
= pThere are no more than 2 children since the next even n+2 has 1-bit count
CountOneBits(n+2) <= CountOneBits(n) + 1
equality when n==0 mod 4, otherwise lessdue to flipping run of 1-bits 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 childful vertices in a given row (those 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 125Options
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 N-1 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).
The default is a directed graph with edges both ways between vertices (like most Graph::Maker directed graphs). This is parameter direction_type => 'both'.
Optional direction_type => 'bigger' or 'child' gives edges directed to the bigger vertex number, so from smaller to bigger. This means parent down to child.
Option direction_type => 'smaller' or 'parent' gives edges directed to the smaller vertex number, so from bigger to smaller. This is from child up to parent.
FUNCTIONS
$graph = Graph::Maker->new ('binary_beanstalk', key => value, ...)-
The key/value parameters are
height => integer N => integer direction_type => string, "both" (default), "bigger", "smaller", "parent, "child" graph_maker => subr(key=>value) constructor, default Graph->newOther parameters are passed to the constructor, either
graph_makerorGraph->new().If the graph is directed (the default) then edges are added as described in "Options" above. Option
undirected => 1is an undirected graph and for it there is always a single edge between parent and child.
HOUSE OF GRAPHS
House of Graphs entries for graphs here include
1310 N=1 (height=1), singleton
19655 N=2 (height=2), path-2
32234 N=3, path-3
500 N=4 (height=3), star-4, claw
30 N=5, fork
334 N=6 (height=4), H graph
714 N=7
502 N=8 (height=5)
60 N=13OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this tree include
http://oeis.org/A179016 (etc)
A011371 parent vertex, n-CountOneBits(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 non-leaf vertices
A179016 trunk vertices
A213712 trunk increments, = count 1-bits 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 non-trunk vertices
A213717 non-trunk non-leaf vertices
A213731 0=leaf, 1=trunk, 2=non-trunk,non-leaf
A213730 start of non-trunk subtree
A213715 trunk position within non-leafs
A213716 non-trunk position within non-leafs
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 non-leaf
A257130 new high positions of nth leaf - nth non-leaf
A218254 paths to root 0
A213707 positions of root 0 in these paths
A218604 num vertices after trunk in row
A213714 how many non-leaf 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^k-1
A213709 depth levels from n=2^k-1 to n=2^(k+1)-1
A213711 how many n=2^k-1 blocks preceding given depth
A213722 num non-trunk,non-leaf v between 2^n <= v < 2^(n+1)SEE ALSO
Graph::Maker, Graph::Maker::BinomialTree
HOME PAGE
http://user42.tuxfamily.org/graph-maker-other/index.html
LICENSE
Copyright 2015, 2016, 2017, 2018, 2019, 2020, 2021 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/.