DATABASE SYSTEM CONCEPTS LCPO
DATABASE SYSTEM CONCEPTS LCPO
7th Edition
ISBN: 9781265586577
Author: SILBERSCHATZ
Publisher: MCG
Expert Solution & Answer
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Chapter 16, Problem 12PE

Explanation of Solution

Proof:

  • Let the number of join orders will be,

Numberofjoinorders=(2n2)!(n1)!

  • A complete binary tree consists of several internal nodes and each internal node contains exactly 2 children.
  • Each join order is considered as a complete binary tree and in which all the relations are considered as the leaves.
  • There is a bijection between the count of binary tress of “n-1” nodes and count of binary trees of “n” leaves when considering a database with “n” relations (leaf nodes).
  • Hence, the total number of created binary trees are,

Numberofcompletebinarytrees=1n(2(n1)(n1))

  • Each complete binary tree of “n” leaves will contain “n-1” nodes.
  • Conversely, add “n” leaves to the binary tree to make this tree to a complete binary tree.
  • The number of binary trees that contains “n-1” nodes is,

Numberofcompletebinarytrees=1n(2(n1

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