. Answer the following with respect to red-black trees. a. Show the red-black trees that result after successively inserting the keys 41, 38, 31, 12, 19, 8 into an initially empty red-black tree. Now show the red-black trees that result from the successive deletion of the keys in the order 8, 12, 19, 31, 38, 41.
1. Answer the following with respect to red-black trees.
a. Show the red-black trees that result after successively inserting the keys 41, 38, 31, 12, 19, 8 into an initially empty red-black tree. Now show the red-black trees that result from the successive deletion of the keys in the order 8, 12, 19, 31, 38, 41.
b. Show the red-black trees that result after successively inserting the keys 41, 38, 31, 12, 19, 8 into an initially empty red-black tree.
c. Show that the longest simple path from a node x in a red-black tree to a descendant leaf has length at most twice that of the shortest simple path from node x to a descendant leaf.
d. Suppose that a node x is inserted into a red-black tree and then is immediately deleted from the tree. Is the resulting red-black tree the same as the initial red-black tree? Justify your answer.
e. What is the largest possible number of internal nodes in a red-black tree with black-height k? What is the smallest possible number?
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