In a left-leaning red-black binary search tree (LLRBBST) with 64 nodes and two paths from the root to a null node P1 and P2, which of the following could be true?

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BlackTom AI · Accepted Answer

This question asks which statement could be true in a left-leaning red-black binary search tree (LLRBBST) with 64 nodes and two root-to-null paths P1 and P2.

First, note the important structural invariant of red-black trees that often carries over to LLRBBSTs: the black-height, i.e., the number of black links on any path from the root to a null leaf, must be the same for all root-to-null paths. Red links can appear along a path, but they do not contribute to the black-height, and the tree is organized so that red links lean left (no right-leaning red link) and there are constraints about consecutive red links.

However, the provided data is incomplete: the question includes an option labeled as (b) but the list of answer options is empty in the JSON. Without the full set of choices, we cannot definitively determine which labeled option could be true. Still, we can discuss the specific scenario described in the given option text to reason about its plausibility in principle.

Option concept discussed (without assuming it is present in the actual list):
- P1 goes through no red links and P2 goes through 1 red link.
- In an LLRBBST, since red links contribute to forming 3-nodes and do not affect black-height, it is possible for two root-to-null paths to differ in the number of red links as long as they have the same number of black links along the path.
- So, a scenario where P1 has zero red links and P2 has exactly one red link could be consistent with equal black-heights, provided the paths contain the same count of black edges from root to the null leaf.
- This reasoning hinges on the existence of a path with one red link that still lands at a null leaf after the same number of black steps as the all-black path. In many red-black configurations, such a balance is possible because red links can be inserted in blocks (as part of 3-nodes) without altering the required black-height.

Now, turning to the argument about why such a statement could be true or false in general:
- Why it could be true: If both P1 and P2 have the same black-height, P2 can include a red link while P1 does not, since red links do not count toward black-height. A tree with 64 nodes could have multiple valid distributions of red links along different root-to-null paths while preserving the invariant.
- Why it could be false: If, in a particular LLRBBST configuration, the specific structural constraints force certain paths to have the same composition of red links or force red links to appear in a way that would make the difference in red-link counts incompatible with equal black-height, then such a deviation (0 red links on one path, 1 red link on another) would not occur.

In summary, the core invariant to consider is black-height equality across root-to-null paths. The idea that one path has 0 red links and another has 1 red link can be compatible with that invariant, but whether this exact scenario can occur depends on the exact arrangement of red and black links in the specific 64-node tree. With the missing full set of options, we cannot definitively label option (b) as could-be-true or not; the determination requires checking the full list and the actual tree structure.

2025W Quiz 4

In a left-leaning red-black binary search tree (LLRBBST) with 64 nodes and two paths from the root to a null node P1 and P2, which of the following could be true?

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2025W Quiz 4

In a left-leaning red-black binary search tree (LLRBBST) with 64 nodes and two paths from the root to a null node P1 and P2, which of the following could be true?

View Explanation

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