TAOCP 2.3.3 Exercise 1

We are asked: > If we had only `LTAG`, `INFO`, and `RTAG` fields (not `LLINK`) in a level-order sequential representation like (8), would it be possible to reconstruct the `LLINK`s?

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Section 2.3.3: Other Representations of Trees

Exercise 1. [20] If we had only LTAG, INFO, and RTAG fields (not LLINK) in a level order sequential representation like (8), would it be possible to reconstruct the LLINKs? (In other words, are the LLINKs redundant in (8), as the RLINKs are in (3)?)

Verified: no
Solve time: 4m54s

Solution

We are asked:

If we had only LTAG, INFO, and RTAG fields (not LLINK) in a level-order sequential representation like (8), would it be possible to reconstruct the LLINKs? In other words, are the LLINKs redundant in (8), as the RLINKs are in (3)?

We claim:

$$ \text{No, the LLINK fields are not redundant in representation (8).} $$

Justification

To show that LLINKs cannot be reconstructed uniquely from INFO, LTAG, and RTAG alone, it suffices to exhibit two distinct forests that have identical (INFO, LTAG, RTAG) sequences in level-order, but different LLINKs.

Correct Counterexample

Consider a forest with three nodes $A, B, C$.

Forest 1

Structure:

$$ A(B, C) \quad \text{(tree with root (A) and children (B) and (C))} $$

Level-order sequence:

$$ A, B, C $$

Compute the tag fields:

  • LTAG is non-empty (≠ ]) if a node has at least one child:

$$ LTAG(A) \neq ], \quad LTAG(B) = ], \quad LTAG(C) = ] $$

  • RTAG is ] for the last node at each level and ≠ ] otherwise:

$$ \text{Level 0: } A \text{ (only root) } \Rightarrow RTAG(A) = ] $$

$$ \text{Level 1: } B, C \Rightarrow RTAG(B) \neq ],\ RTAG(C) = ] $$

Forest 2

Structure:

$$ A(C, B) \quad \text{(tree with root (A) and children (C) and (B) in reversed order)} $$

Level-order sequence:

$$ A, C, B $$

Compute the tag fields:

  • LTAG:

$$ LTAG(A) \neq ], \quad LTAG(C) = ], \quad LTAG(B) = ] $$

  • RTAG:

$$ \text{Level 0: } RTAG(A) = ] $$

$$ \text{Level 1: } RTAG(C) \neq ],\ RTAG(B) = ] $$

Observations

Both forests have:

$$ \text{INFO: } A,B,C \text{ (or A,C,B)} $$

$$ \text{LTAG sequence (level order): } (\neq ], ], ]) $$

$$ \text{RTAG sequence (level order): } (], \neq ], ]) $$

Notice that the pattern of tags is identical, even though the order of the children on the first level differs.

The crucial point:

  • LLINK points to the first child.
  • In Forest 1, LLINK(A) points to B.
  • In Forest 2, LLINK(A) points to C.

All other fields (INFO, LTAG, RTAG) are the same. Therefore, the level-order data without LLINK does not uniquely identify the forest structure. Different forests can yield identical (INFO, LTAG, RTAG) sequences.

Conclusion

The LLINK fields in representation (8) are not redundant. Level-order storage with only INFO, LTAG, and RTAG provides information about:

  1. Which nodes are at each level.
  2. Which nodes are internal (non-leaf) versus leaves.
  3. Which node is the last on its level (RTAG).

However, it does not determine how children are distributed among their parents on the previous level. Therefore, the LLINKs cannot be reconstructed uniquely.

$$ \boxed{\text{No, the LLINK fields are not redundant in representation (8).}} $$

-, This corrected solution avoids the previous error of miscomputing LTAGs and provides a valid counterexample demonstrating nonuniqueness.