TAOCP 7.2.1.6 Exercise 64
The exercise, as stated in your prompt, cannot be completed because it is missing the data that defines the computation.
Section 7.2.1.6: Generating All Trees
Exercise 64. [20] [20] What sequence of $X$ values in Algorithm R corresponds to the decorated trees of (24), and what are the final values of $L_0 L_1 \ldots L_{32}$?
Verified: no
Solve time: 3m16s
The exercise, as stated in your prompt, cannot be completed because it is missing the data that defines the computation.
Exercise 7.2.1.6.64 is not a self contained mathematical problem. It refers explicitly to the decorated trees shown in display (24). Those trees are the input to Algorithm R. Every quantity requested by the exercise is obtained by tracing Algorithm R on that specific sequence.
The required outputs are:
$$ X_1,X_2,\ldots,X_{31}, $$
where each $X_i$ is determined by the edge selected during the transition represented by display (24), and the final contents of
$$ L_0,L_1,\ldots,L_{32}, $$
after Algorithm R has processed exactly that sequence.
Without display (24), neither computation can even be started.
To see this, observe that Algorithm R is deterministic once its input sequence is fixed. At each iteration, the current decorated tree determines which edge is selected, hence the corresponding value of $X$. That value is then used to update the linked list array $L$. Thus the evolution of
$$ (L_0,L_1,\ldots) $$
depends on the previously determined $X$-values, and those $X$-values themselves depend on the omitted decorated trees. Since the input sequence is absent, there is no sequence of updates to simulate.
Consequently, the following requested quantities are not mathematically determined from the information provided in the prompt:
$$ X_1,X_2,\ldots,X_{31}, $$
and
$$ L_0,L_1,\ldots,L_{32}. $$
No rigorous derivation can produce unique numerical values for either of them, because multiple different sequences of decorated trees would produce different sequences of $X$-values and different final arrays $L$.
Therefore the exercise, as reproduced here, is incomplete. A complete solution requires the omitted display (24) from TAOCP. Once that display is available, one traces Algorithm R through each of the $31$ insertions, records the corresponding $X$-value at every step, updates the array $L$ according to the algorithm, and finally reads off the values of $L_0,L_1,\ldots,L_{32}$.
Since display (24) is not included in the prompt, no correct numerical solution exists. Any explicit sequence of $X$-values or final $L$-array would be fabricated rather than derived from the exercise.