Chapter 13 Exercises

We use the following expression S(T) to show a tree T:
if T is null, then S(T) is a null string
if T contains only a root node, S(T) = root
otherwise, let L and R are T's left and right subtrees, respectively, S(T) = root(S(L),S(R))

Exercises
13.1-7 Describe a red-black tree on n keys that realizes the largest possible ratio of red internal nodes to black internal nodes. What is the ratio? What tree has the smallest possible ratio, and what is the ratio?
Largest: Tree: 2(1,3), when 1,3 are red. The ratio is 2.
Smallest: a tree with only a root node. The ratio is 0.

13.2-5 We say that a binary search tree T1 can be right-converted to binary search tree T2 if it is possible to obtain T2 from T1 via a series of calls to RIGHT-ROTATE. Give an example of two trees T1 and T2 such that T1 cannot be right-converted to T2. Then show that if a tree T1 can be right-converted to T2, it can be right-converted using O(n^2) calls to RIGHT-ROTATE
The counter-example is : T1: 1(,2), T2: 2(1,). Or, a simple case that T2 can be right-converted to T1. It can be proved that when T2 can be right-converted to T1, T1 is not so to T2.
The O(n^2) proof is also not difficult. If T1's root is different from T2's, O(n) RIGHT-ROTATEs are needed to turn T1's root to T2's. Then, think of the right subtrees of T1's and T2's. We can conclude a recurrence of T(n) = T(n-1) + O(n), thus T(n) = O(n^2)

13.4-7 Suppose that a node x is inserted into a red-black tree with RB-INSERT and then immediately deleted with RB-DELETE. Is the resulting red-black tree the same as the initial red-black tree? Justify your answer.
The answer is no. Consider the tree T = 2(1,3), where 1 and 3 are the two red nodes. When we insert 1.5, which is initially inserted as 1's right node, 1 and 3 nodes will turn black, and 1.5 will be red. Deleting 1.5 node doesn't effect on other nodes. Thus, after inserting and deleting 1.5, the 1 and 3 nodes will turn black.
Another example is the Figure 13.4 on the book. The procedure of inserting 4 is shown on Figure 13.4, and deleting 4 also has no effect on other nodes. The resulting tree is also different from Figure 13.4(a).

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