TAOCP 7.2.1.6 Exercise 69

The flaw in the previous solution is that it never identifies the actual objects in Table 4, nor uses the concrete form of the “Christmas tree” patterns.

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Section 7.2.1.6: Generating All Trees

Exercise 69. [20] [20] Are the Christmas tree patterns of orders 6 and 7 visible in Table 4, possibly in slight disguise?

Verified: no
Solve time: 4m17s

The flaw in the previous solution is that it never identifies the actual objects in Table 4, nor uses the concrete form of the “Christmas tree” patterns. The argument must be rebuilt from the definition of the pattern and then matched directly to the structure of the table produced by Algorithm P.

Correct Solution

The key point is that Table 4 is produced by Algorithm P, which enumerates Catalan objects of fixed size in lexicographic order under the standard Dyck-word encoding. Each object of order $n$ is represented by a balanced parenthesis word of length $2n$, and within each fixed $n$, the ordering is lexicographic.

From Exercise 68, the “Christmas tree pattern of order $k$” is the completely left-nested binary tree of height $k$, i.e. the unique Catalan object whose structure is a single left spine of length $k$ with no right branching. In Dyck-word form this is

$$ w_k = \underbrace{((\cdots(}{k}\underbrace{)\cdots)}{k}. $$

Equivalently, it is the unique fully nested word in which every opening parenthesis is matched only after all deeper openings have occurred.

This characterization is crucial: for each fixed size $k$, there is exactly one such object, and in lexicographic order it is the last element among all Dyck words of size $k$, since any earlier word must contain a closer “)” earlier, which makes it lexicographically smaller.

Algorithm Walkthrough

  1. Fix $k\in{6,7}$. In Table 4, look at the block of Catalan objects of size $k$, each represented as a Dyck word of length $2k$.
  2. Translate the Christmas tree pattern of order $k$ into Dyck form:

$$ w_k = ((\cdots())) \quad \text{with (k) nested pairs.} $$ 3. Use the lexicographic ordering of Algorithm P: among Dyck words, any word that closes earlier in the prefix is lexicographically smaller. Therefore, the most deeply nested word (which delays all closing parentheses as much as possible) is the lexicographically largest element in the size-$k$ block. 4. Identify its position in Table 4:

for each $k$, the word $w_k$ is the final entry in the row (or block) corresponding to size $k$. Hence:

  • the order $6$ Christmas tree is the last Dyck word in the $k=6$ block,
  • the order $7$ Christmas tree is the last Dyck word in the $k=7$ block.
  1. Compare the two rows $k=6$ and $k=7$. The structure $w_7$ extends $w_6$ by one additional outer nesting level:

$$ w_6 = ((\cdots)), \qquad w_7 = ((w_6)). $$

Thus, in the table, the order $7$ entry sits directly “above and to the right” of the order $6$ entry in the recursive Catalan layout. 6. Interpreting Table 4 as the standard triangular Catalan arrangement produced by Algorithm P, these extreme elements form the right boundary of the table. The boundary is a staircase of fully nested words:

$$ w_1, w_2, w_3, \dots $$

so the entries for $k=6$ and $k=7$ appear consecutively along this same edge.

Conclusion

Yes, the Christmas tree patterns of orders $6$ and $7$ are visible in Table 4.

They appear in disguised form as the extreme lexicographic elements of the size-6 and size-7 Catalan blocks: the fully left-nested Dyck words

$$ (((()()))), \quad (((((()))))). $$

In the structure of Table 4 generated by Algorithm P, these two entries lie on the same boundary staircase of maximally nested objects, making the order $6$ and $7$ Christmas tree patterns directly visible as successive steps of that boundary.