Proof that a subtree of a red-black tree has no more than $\frac{3n}{4}$ nodes

Question

I have a red-black tree with $n$ nodes, rooted at $x$. How can I prove or disprove that the number of nodes in any subtree of $x$ (including the root of the subtree) will never be greater than $\frac{3n}{4}$?

Answer 1

Here is a natural approach to explore the extreme case.

Let node $r$ be the root of a red-black tree.

Let its left subtree be a red-black tree of black height $h$ that has the minimum number of the nodes among red-black tree of the same black height.

Let its right subtree be a red-black tree of the same black height $h$ that has the maximum number of the nodes among red-black tree of the same black height.

Now let $h=1,2,3,\cdots$ be larger and larger. Can you see the trend?

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Here is a related exercise.

Exercise (red-black trees may not be weight-balanced). Given $\alpha\lt 1$, show that there is a red-black tree which has a proper subtree that has more than $\alpha$ of all nodes.