Graph::Maker::BinaryBeanstalk - create binary beanstalk graph
use Graph::Maker::BinaryBeanstalk; $graph = Graph::Maker->new ('binary_beanstalk', height => 4);
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 1-bits so parent 9-2=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 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 = p
There 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 less
due 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 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
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.
N are given then the tree stops at whichever
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).
$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::BalancedTree, if the graph is directed (the default) then edges are added both up and down between each parent and child. Option
undirected => 1creates an undirected graph and for it there is a single edge between parent and child.
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 (path-2)
- N=3, <https://hog.grinvin.org/ViewGraphInfo.action?id=32234> path-3
- 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
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this tree include
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)
Copyright 2015, 2016, 2017, 2018, 2019 Kevin Ryde
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