Given any perfectly balanced binary search tree, is it always possible to assign a coloring to the nodes so that it becomes a Red-Black tree? If so, how do you prove this, and if false, what would be a counterexample?
Recall the following definitions.
Perfectly balanced node: A node of a binary tree is said to be perfectly balanced if the absolute difference in the number of vertices in its left subtree and its right subtree is atmost 1.
Perfectly Balanced Tree: A binary tree is called perfectly balanced if all its nodes are perfectly balanced.
Trivial Fact: The height of a perfectly balanced tree is $\Theta (\log n)$ where $n$ is the number of nodes in the tree.