RBT_RTT_11
Consider invalid Red Black Tree below.
A___ rotation at node_____yields a valid red-black tree.

Question

BlackTom AI · Accepted Answer

Question restatement: The problem asks which rotation would Fix an invalid Red-Black Tree, specifying a left rotation at a particular node labeled 40 to be performed at a node y.

Option 1: left, 60. This suggests performing a left rotation at the node labeled 60. If the actual imbalance or red-black property violation occurs around a different node than 60, or if rotating at 60 would propagate the violation to another part of the tree, this choice would fail to restore the required properties. Moreover, the specific pair (left, 60) may not address the parent-child relationships needed to re-color and restructure the tree correctly.

Option 2: no rotation is needed. If the tree is indeed invalid due to a local structural violation (for example, a red-red violation or black-height mismatch), skipping rotation would leave the properties violated. Red-black trees rely on rotations as a key mechanism to fix violations while preserving the binary search tree order and the black-height constraints. Therefore, this option is unlikely to yield a valid tree given an invalid configuration.

Option 3: left, 40. This option indicates performing a left rotation at the node labeled 40. If the invalidity stems from a right-leaning or left-right heavy subtree around 40, a left rotation at that node can rebalance pointers and, with potential recoloring, restore both the BST structure and the red-black properties. The choice of 40 as the rotation node aligns with scenarios where a local rotation at that specific node corrects the violation without cascading into deeper structural issues.

Option 4: left, 20. Rotating at node 20 could be appropriate in a different local configuration, but if the actual violation is centered around node 40 or its parent, rotating at 20 may not fix the red-red relationship or the black-height balance. It risks leaving the tree in an invalid state or introducing new imbalances that violate red-black rules.

Summary of rationale against incorrect options: Option 1 targets the wrong pivot and may not resolve the root cause of the imbalance. Option 2 assumes no rotation is necessary despite an invalid tree, which contradicts the need to restore red-black properties via rotations. Option 4 rotates at a different node than the one implicated by the problem’s setup, likely failing to rectify the specific violation.

Supporting intuition for the correct approach: In many red-black repair procedures, selecting the precise rotation node to address a local structural fault is essential to re-establish both BST order and color constraints with minimal changes. Rotating at 40, when the setup indicates that is the impactful pivot, can realign children, grandparent relations, and colors to satisfy all red-black properties after a single corrective step.

RBT_RTT_11 Consider invalid Red Black Tree below. A_ rotation at node___yields a valid red-black tree.

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RBT_RTT_11 Consider invalid Red Black Tree below. A___ rotation at node_____yields a valid red-black tree.