Imagine a red-black tree. Is there always a sequence of insertions and deletions that creates it?

Question

Let's assume the following definition of a red-black tree:

1. It is a binary search tree.
2. Each node is colored either red or black. The root is black.
3. Two nodes connected by an edge cannot be red at the same time.
4. Here should be a good definition of a NIL leaf, like on wiki. The NIL leaf is colored black.
5. A path from the root to any NIL leaf contains the same number of black nodes.

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Question

Suppose that you have implemented the insert and delete operations for the red-black tree. Now, if you are given a valid red-black tree, is there always a sequence of insert and delete operations that constructs it?

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Motivation

This question is motivated by this question and by the discussion from this question.

Personally, I do believe that if you imagine a valid red-black tree consisting only of black nodes (which implies that you are imagining a perfectly balanced tree), there is a sequence of insert and delete operations that constructs it. However,

1. I do not know how to accurately prove that
2. I am also interested in the more general case

Answer 1

The insert and delete operations in a red-black tree include the balancing needed to maintain the red-black properties.

The problem with non(left- or right-) leaning red black trees is that there are multiple ways to restore the red-blackness after the basic delete or insert.
It is not the insert or the delete that transforms the tree, but the rebalancing and rotation that happens afterwards to preserve/restore the red blackness of the tree.

The basic description of the red-black tree does not prescribe which of the possible routes to take.
It may not be possible to figure out how to exactly reconstruct a given red black tree, because the rebalancing need not be deterministic.

This has been 'solved' with left-leaning red black trees.
There is only one way the balancing is done. So any given leaning red black tree can be reconstructed using inserts and deletes, because the rebalancing/rotations are done in a specific deterministic way.

This does not mean that left-leaning RB-trees are better or more efficient, what they gain on one hand by using deterministic balancing rules, they lose on the other by more complex balancing code.

As per @Anton's comment:
There is an algorithm which uses the operation insert and delete to construct a valid red-black tree consisting only of black nodes. It uses $(h+2)⋅2^{h}−1$ insertions/deletions to create a tree of height $h$. First, we can create a perfectly balanced red-black tree in breadth-first manner using $2^{h+1}−1$ insertions, then using $h∗2^{h-1}$ insertions and the same amount of deletions repaint it into a completely black tree. The trick here is to move up $h$ times the lowest red layer up the tree until it reaches the root.

I think a complete balancing algorithm like Day-Stout-Warren would be more efficient though.