Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

Figure 10.27: Correspondence between a (2,4) tree
and a red-black tree: (a) 2-node; (b) 3-node; (c) 4-node.
The correspondence between (2,4) trees and red-black trees provides important
intuition that we will use in our discussion of how to perform updates in red-black
trees. In fact, the update algorithms for red-black trees are mysteriously complex
without this intuition.
Proposition 10.9: The height of a red-black tree storing n entries is O(logn).
Justification: Let T be a red-black tree storing n entries, and let h be the
height of T. We justify this proposition by establishing the following fact:
log(n + 1) ≤ h ≤ 2log(n + 1).
Let d be the common black depth of all the external nodes of T. Let T ′ be the (2,4)
tree associated with T, and let h ′ be the height of T ′. Because of the correspondence
between red-black trees and (2,4) trees, we know that h ′ = d. Hence, by Proposition
10.8, d = h ′ ≤ log(n + 1). By the internal node property, h ≤ 2d. Thus, we obtain h ≤
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