Draw the initial query tree for each of these queries, then show how the query tree is optimized (one rule at a time) by the algorithm outlined in chapter 19 and write the SQL query for the final optimized query tree.

Q1: List the lastnames of pair of authors who co-authored at least one book.

SELECT a1.AuthorLast, a2.AuthorLast FROM Author a1, Wrote w1, Author a2, Wrote w2 WHERE a1.AuthorNum = w1.AuthorNum and a2.AuthorNum = w2.AuthorNum and w1.BookCode = w2.BookCode and a1.AuthorNum < a2.AuthorNum;