(a) Add successive nodes with keys 48.8.69. 7.50. 52. 71. 1/23/10 into an empty binary search tree. Document the condition after each insertion operation. Is the resulting tree a red black tree? If yes, color it, if no, briefly explain why. (b) Let a, b and c be arbitrary nodes from the respective subtrees A, B and C' in the following binary search tree T: y Next are d.. d. d. the depths of the three nodes. Determine the new depths of the three nodes using the rotateRight(T,) and rotateLeft(T,t) operations. 05-4/5 (c) Let and y be arbitrary nodes in a binary search tree T. Prove or disprove the following statement: The following applies: delete(delete(T, x), y) = delete(delete(T, y), x), i.e. the result of two deletion operations is independent of their order. (d) Let T be a binary search tree with pairwise different values and u be any node in T with two children Show that the predecessor (next smaller value) and successor (next larger value) of v are half-leaves or leaves must be. Note: Can the predecessor have a right child or the successor a left child? (e) Argue that any comparison-based algorithm for constructing a binary search tree from an arbitrary list of n elements takes N(n log n) time in the worst case.
(a) Add successive nodes with keys 48.8.69. 7.50. 52. 71. 1/23/10 into an empty binary search tree. Document the condition after each insertion operation. Is the resulting tree a red black tree? If yes, color it, if no, briefly explain why. (b) Let a, b and c be arbitrary nodes from the respective subtrees A, B and C' in the following binary search tree T: y Next are d.. d. d. the depths of the three nodes. Determine the new depths of the three nodes using the rotateRight(T,) and rotateLeft(T,t) operations. 05-4/5 (c) Let and y be arbitrary nodes in a binary search tree T. Prove or disprove the following statement: The following applies: delete(delete(T, x), y) = delete(delete(T, y), x), i.e. the result of two deletion operations is independent of their order. (d) Let T be a binary search tree with pairwise different values and u be any node in T with two children Show that the predecessor (next smaller value) and successor (next larger value) of v are half-leaves or leaves must be. Note: Can the predecessor have a right child or the successor a left child? (e) Argue that any comparison-based algorithm for constructing a binary search tree from an arbitrary list of n elements takes N(n log n) time in the worst case.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Transcribed Image Text:(a) Add successive nodes with keys 48.8.69. 7.50. 52. 71. 1/23/10 into an empty binary search
tree. Document the condition after each insertion operation. Is the resulting tree
a red black tree? If yes, color it, if no, briefly explain why.
(b) Let a, b and c be arbitrary nodes from the respective subtrees A, B and C' in the following binary search tree T:
y
Next are d.. d. d. the depths of the three nodes. Determine the new depths of the three nodes using the
rotateRight(T,) and rotateLeft(T,t) operations.
05-4/5
(c) Let and y be arbitrary nodes in a binary search tree T. Prove or disprove the following statement:
The following applies: delete(delete(T, x), y) = delete(delete(T, y), x), i.e. the result of two deletion operations
is independent of their order.
(d) Let T be a binary search tree with pairwise different values and u be any node in T with two children
Show that the predecessor (next smaller value) and successor (next larger value) of v are half-leaves
or leaves must be.
Note: Can the predecessor have a right child or the successor a left child?
(e) Argue that any comparison-based algorithm for constructing a binary search tree from an
arbitrary list of n elements takes N(n log n) time in the worst case.
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