Suppose that we "absorb" every red node in a red-black tree into its black parent, so that the children of the red node become children of the black parent. (Ignore what happens to the keys.) What are the possible degrees a of black node after all its red children are absorbed? What can you say about the depths of the leaves of the resulting tree?
Two things are clear from the properties:
- Red nodes have black children.
- Red nodes have black parents.
Both follow from property 4, which implies that a red node cannot be a parent of another red node. Alternatively put, each path to a leaf may go through subsequent black nodes, but no two red nodes in a row.
This means that the most complicated subtree is going to look like this:
If we "absorb" the red nodes, node 4 will end up having four children (1, 3, 5, 7), that is, a degree of $4$ and no more.
The depth of the resulting leaves can at most halve. That is, if a leaf had depth $n$, it's new depth is going to be at least $\lceil n / 2 \rceil$.