Let's say we have a red-black tree of $n$ total nodes where all keys are distinct. The subtree rooted at the root node's left child has $n_L$ nodes, and similarly the subtree rooted at the root node's right child has $n_R$ nodes.
What is the range of values on $n_L$ and $n_R$ for any $n \geq 3$?
From Red Black Tree vs AVL Tree
For example, in the above image: $n = 8, n_L = 1, n_R = 6$.
Let $b$ be the black height of the left subtree, which is also the black height of the right subtree.
The number of nodes in the left subtree is at least $2^{b}-1$. $2^b-1$ is reached when the left subtree is a perfect binary tree of black nodes and height $b-1$.
The number of nodes in the right subtree is at most $2^{2b+1}-1$. $2^{2b+1}-1$ is reached when the root is black and the right subtree is a perfect binary tree of height $2b$ with nodes colored red, black, red, black and so on alternatively from top to bottom.
$$2^b-1 + 2^{2b+1}-1\ge n-1$$
Solving for $b$, we obtain $$ b\ge f(n) := \lceil \log_2(\sqrt{8n+9}-1)\rceil-2$$
So, the range for $n_L$ is $[2^{f(n)}-1, n - 2^{f(n)}]$, which is, by symmetry, also the range for $n_R$.
For example, when $n=8$, $f(n)=1$, the range is $[1, 6]$ as illustrated in the question.
