The General Predecessor Tree
The Binary Predecessor Tree
The State Diagram for the Left Descent Assemblies
The State Diagram for the Even Collatz Iterates
The State Diagram for Chained Left Descent Assemblies
The Abstract Tree -- Recursive Definition
divide node contents into 3 equal subsets: {0 mod 3} {2 mod 3} {1 mod 3} leaves | s process | b process V V {integers equally in {integers equally in {0, 1, 2} mod 3} {0, 1, 2} mod 3}
Figure 3: One Cycle of the Iterative Development of the Abstract Generation Tree
The Abstract Tree to 3 Levels of Depth
e / {8n+5} \
/ \
/ \
/ \
s b
/ \ / \
/ \ / \
ss sb bs bb
/ \ / \ / \ / \
sss ssb sbs sbb bss bsb bbs bbb
/ \ / \ / \ / \ / \ / \ / \ / \
Figure 4: First Three Levels of Abstract Predecessor Tree Using Symbolic Set Names
Diagram Showing Set Size Decrease During Abstract Tree Growth
Schematic of Binary Tree Construction Using L.d.a.s and Extensions
First of Two Steps in Derivation of c[d] Values for Sets Within L.d.a.s
First of Two Steps in Derivation of c[d] Values for Sets Within L.d.a.s
The table entries are developed only through three levels of growth, corresponding to the depth in Figure 4. The prefixed e is included in the set names to indicate that the envisioned root is a member of the extended set of parents of left descents. An sss sequence not rooted in the e set will be a set (16n+15) containing 4 times the number of elements as the anchored set developed here for the esss sequence (64n+15). Each dn+c formula denotes the set of odd integers congruent to c mod d.
Allowed Applied Resulting Terminating
From Subset Move Transform Result Set Name Set Names
e 24n+5 s (2(24n+5)-1)/3 16n+3 es
24n+13 b (4(24n+13)-1)/3 32n+17 eb
24n+21 null et
es 48n+3 null est
48n+19 b (4(48n+19)-1)/3 64n+25 esb
48n+35 s (2(48n+35)-1)/3 32n+23 ess
eb 96n+17 s (2(96n+17)-1)/3 64n+11 ebs
96n+49 b (4(96n+49)-1)/3 128n+65 ebb
96n+81 null ebt
esb 192n+25 b (4(192n+25)-1)/3 768n+33 esbb
192n+89 s (2(192n+89)-1)/3 384n+59 esbs
192n+153 null esbt
ess 96n+23 s (2(96n+23)-1)/3 64n+15 esss
96n+55 b (4(96n+55)-1)/3 128n+73 essb
96n+87 null esst
ebs 192n+11 s (2(192n+11)-1)/3 128n+7 ebss
192n+75 null ebst
192n+139 b (4(192n+139)-1)/3 256n+185 ebsb
ebb 384n+65 s (2(384n+65)-1)/3 256n+43 ebbs
384n+193 b (4(384n+193)-1)/3 512n+257 ebbb
384n+321 null ebbt
Figure 5: Stepwise Development of Abstract Generation Tree Rooted in Extended Parents , Showing Set Contents
Second of Two Steps in Derivation of c[d] Values for Sets Within L.d.a.s
path leaf set immediate parent grandparent great grandparent
et 2^3*3^1*n+21
ebt 2^5*3^1*n+81 2^3*3^2*n+61
est 2^4*3^1*n+3 2^3*3^2*n+5
ebbt 2^7*3^1*n+321 2^5*3^2*n+241 2^3*3^3*n+181
ebst 2^6*3^1*n+75 2^5*3^2*n+113 2^3*3^3*n+85
esbt 2^6*3^1*n+153 2^4*3^2*n+115 2^3*3^3*n+173
esst 2^5*3^1*n+87 2^4*3^2*n+131 2^3*3^3*n+197
ebbbt 2^9*3^1*n+1281 2^7*3^2*n+961 2^5*3^3*n+721 2^3*3^4*n+541
ebbst 2^8*3^1*n+555 2^7*3^2*n+833 2^5*3^3*n+625 2^3*3^4*n+469
ebsbt 2^8*3^1*n+441 2^6*3^2*n+331 2^5*3^3*n+497 2^3*3^4*n+373
Figure 6: Tracing upward from terminating paths to develop integer sets completely determined in context
An Annotated Pascal Triangle
The Table of Cardinalities of d=2a3b values of c[d] for l.d.a. elements
Due to its size, this file is reached by reference only. Results Derived From a Very Large Predecessor Tree Computer Generation Run
The Three Graphics Elements Available for Binary Predecessor Tree Construction
Results of Beginning the Binary Predecessor Tree Construction
