We are concerned with the odd positive integers with respect to their congruences mod 3 and mod 8. The value mod 3 determines whether an odd integer is a leaf node in the Collatz predecessor tree or is the parent of an 's' or a 'b' step. For a 'b' step (from any 1[3]), the Collatz predecessor is calculated using (4*n-1)/3 and for an 's' step (from any 2[3]), the Collatz predecessor is calculated using (2*n-1)/3. Since we consider only the odd integers, there are just four values mod 8 to look at: 1[8], 3[8], 5[8], and 7[8]). We are not concerned with leaf nodes (i.e. all 0[3]) because leaf nodes do not head l.d.a.s and always terminate l.d.a.s. A leaf node is the only element which can terminate an l.d.a. Thus only eight states need be shown, two (1[3] and 2[3]) mod 3, in combination with the above four mod 8.
The state transition diagram doesn't show the transition from the penultimate element of an l.d.a. to the leaf node; including that transition would just involve adding an arrow from each node in the diagram pointing to a collective leaf node. The l.d.a.s which consist of a single header/leaf, e.g. 21[24] also are not depicted. In principle, an l.d.a. can continue indefinitely, but as the length of an l.d.a. increases its density of occurence rapidly decreases so lengthy l.d.a.s are relatively rare.
The odd integers 5[8] which start (head) l.d.a.s are shown as the red nodes A and B in the state diagram. Working through all possible transitions from each possible state allows construction of the complete transition diagram. The following table provides the basis for the edges in the state transition diagram.
from step results
A 1[3] 5[8] b --> C 1[3] 1[8] and D 2[3] 1[8]
B 2[3] 5[8] s --> E 1[3] 3[8] and F 2[3] 3[8]
C 1[3] 1[8] b --> C 1[3] 1[8] and D 2[3] 1[8]
D 2[3] 1[8] s --> E 1[3] 3[8] and F 2[3] 3[8]
E 1[3] 3[8] b --> C 1[3] 1[8] and D 2[3] 1[8]
F 2[3] 3[8] s --> G 1[3] 7[8] and H 2[3] 7[8]
F 2[3] 1[8] s --> G 1[3] 7[8] and H 2[3] 7[8]
G 1[3] 7[8] b --> C 1[3] 1[8] and D 2[3] 1[8]
H 2[3] 7[8] s --> G 1[3] 7[8] and H 2[3] 7[8]
Note that all the other possible states are reached from one or the other of the header states, but nodes A and B are never reached from within an l.d.a. (This point is already known -- see Margaret Maxfield's proof among these web pages.) The fact that there are no edges reaching an extension/header from within the diagram supports my claim that l.d.a.s are a useful unit in dissecting the Collatz predecessor graph for further analysis. This is a key fact.
As an example, we might take the l.d.a. from 445 to 27, which runs 445, 593, 263, 175, 233, 155, 103, 137, 91, 121, 161, 107, 71, 47, 31, 41, 27. This l.d.a., which appears among low integers, starts the Collatz trajectory from 27 which is often cited to show the caprice of the Collatz sequence. This l.d.a. proceeds through
A(445), b->D(593), s->F(395), s->H(263), s->G(175), b->D(233), s->F(155), s->G(103), b->D(137), s->E(91), b->C(121), b->D(161), s->F(107), s->H(71), s->H(47), s->G(31), b->D(41), s->leaf(27),where the last step is not on the diagram because it goes to the terminating leaf node. Naturally, since it is started via A->D, the alternative starting edges A-->C, B-->E, and B-->F cannot appear in this trajectory, but of the other edges all except C-->C, E-->D, and G-->C are traversed in this l.d.a. A more detailed exhibition of this same l.d.a. which includes the set contents at each stage is available.
