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Problem 1. For cach integer n >1 we define a tree Tn, recursively, as follows. Tị and T2 consist of only a single node. For n 2 3, T, is obtained from three copies of Tn/3, one copy of Tn/3, and three additional nodes, by connecting them as follows: Tim3 Notations [r] and [x] represent the floor and ceiling functions; the first one rounds a real number I to the largest integer a < r and the second one rounds I to the smallest integer b > 1. Let t(n) be the number of nodes in Tn. (a) Give a recurrence equation for t(n) and justify it. (b) Draw T19. (You can use a drawing software or draw it by hand, and include a pdf file in the latex source.) (c) Give the asymptotic formula for t(n), by using Master Theorem to solve the recurrence from part. (a), Justify vour solution.

Problem 1. For cach integer n >1 we define a tree Tn, recursively, as follows. Tị and T2 consist of only a single node. For n 2 3, T, is obtained from three copies of Tn/3, one copy of Tn/3, and three additional nodes, by connecting them as follows: Tim3 Notations [r] and [x] represent the floor and ceiling functions; the first one rounds a real number I to the largest integer a < r and the second one rounds I to the smallest integer b > 1. Let t(n) be the number of nodes in Tn. (a) Give a recurrence equation for t(n) and justify it. (b) Draw T19. (You can use a drawing software or draw it by hand, and include a pdf file in the latex source.) (c) Give the asymptotic formula for t(n), by using Master Theorem to solve the recurrence from part. (a), Justify vour solution.

Database System Concepts
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
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This is a question about Divide-and-Conquer Recurrence Equations. 

Can you show me the answer with clear explanations for questions a and c? Thank you.

Problem 1. For each integer n > 1 we define a tree Tn, recursively, as follows. T1 and T2 consist of only a single node. For n 2 3, T, is obtained from three copies of Tn/3, one copy of Tn/3, and three additional nodes, by connecting them as follows: Notations [r] and [x] represent the floor and ceiling functions; the first one rounds a real number r to the largest integer a <r and the second one rounds r to the smallest integer b> r. Let l(n) be the number of nodes in Tn- (a) Give a recurrence equation for t(n) and justify it. (b) Draw T19. (You can use a drawing software or draw it by hand, and include a pdf file in the latex source.) (c) Give the asymptotic formula for t(n), by using Master Theorem to solve the recurrence from part (a). Justify your solution.
Transcribed Image Text:Problem 1. For each integer n > 1 we define a tree Tn, recursively, as follows. T1 and T2 consist of only a single node. For n 2 3, T, is obtained from three copies of Tn/3, one copy of Tn/3, and three additional nodes, by connecting them as follows: Notations [r] and [x] represent the floor and ceiling functions; the first one rounds a real number r to the largest integer a <r and the second one rounds r to the smallest integer b> r. Let l(n) be the number of nodes in Tn- (a) Give a recurrence equation for t(n) and justify it. (b) Draw T19. (You can use a drawing software or draw it by hand, and include a pdf file in the latex source.) (c) Give the asymptotic formula for t(n), by using Master Theorem to solve the recurrence from part (a). Justify your solution.
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