{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW6-solutions

# HW6-solutions - EECS 203 DISCRETE MATHEMATICS Homework 6...

This preview shows pages 1–2. Sign up to view the full content.

EECS 203: DISCRETE MATHEMATICS Homework 6 Solutions 1. (4 points) A rooted ternary tree is either a single vertex (which is the root) or consists of three ternary trees T 1 ,T 2 ,T 3 with roots r 1 ,r 2 ,r 3 , a new root vertex r , and edges { r,r 1 } , { r,r 2 } , { r,r 3 } . What is the maximum number of vertices in a ternary tree with height h ? Your proof should be by induction. Solution: Let V ( n ) be the maximum number of vertices in a ternary tree with height n . Using the recursive deﬁnition of ternary tree we have: V (0) = 1 V ( n ) = 3 V ( n - 1) + 1 for n > 0 In order to make an educated guess at what T ( n ) might be, note that the maximum number of vertices at height 0 is 1, at height 1 is 3, and in general, at height h the maximum number of vertices is 3 h . A good guess is that V ( n ) = 3 0 + 3 1 + ··· + 3 n = 3 n +1 - 1 2 . We prove this by induction. Base case: V (0) = 1 = 3 1 - 1 2 . General case: n > 0 Assume inductively that V ( n 0 ) = 3 n 0 +1 - 1 2 for all n 0 < n . Then: V ( n ) = 3 V ( n - 1) + 1 { def of V } = 3( 3 n - 1 2 ) + 1 { inductive assumption } = 3 n +1 - 3 2 + 1 = 3 n +1 - 1 2 { algebra } 2. (4 points) The sequence ( a 0 ,a 1 ,... ) is deﬁned recursively as follows: a 0 = 0 and, for n > 0, a n = ± a n/ 2 for n even 1 + a ( n - 1) / 2 for n odd Prove by induction that a n is the number of 1s in the binary representation of n . Solution: Let ( b m b m - 1 ...b 1 b 0 ) 2 be the binary representation of n (each b i is a bit). Note that (i) if n is odd then b 0 = 1 and n - 1 = ( b m ...b 1 0) 2 and (ii) if n is even then b 0 = 0 and n/ 2 = ( b m ...b 1 ) 2 . Proof by induction: The base case is

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

HW6-solutions - EECS 203 DISCRETE MATHEMATICS Homework 6...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online