2011Winter2009_Assignment3_solution

# 2011Winter2009_Assignment3_solution - Department of...

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1 Department of Computer Science and Engineering York University, Winter 2009 CSE 2011: Assignment 3 Due Date - Saturday, May 23, 7pm! Question 1 3-ary Tree [35 points] Let T be a full 3-ary tree (see Figure 1); that is, each parent has exactly 3 children. Let H(n) denote the height of this tree, where n represents the number of nodes in the tree. Prove the following is true for all full 3-ary trees: H(n) = log 3 (2 n+1) -1 Hint: use induction! Figure 1 Full 3-are tree Answer: Use induction. Case: n=1 H(1) = log 3 (2*1+1) – 1 = 0 - OK! Case: n=2 CANNOT BE! We cannot have 2 nodes in 3-ary tree. New valid n: When adding new set of nodes (at n+1 level), every node at level 1 will now become the root of a sub-tree that is identical to the entire tree of step n. Hence, n new = 3*n prev + 1 n prev = (n new -1)/3 Case: n=4 H(4) = log 3 (2*4+1) – 1 = 1 - OK! Case: n H(n) = H(n prev ) + 1 = = log 3 (2*n prev +1) -1 + 1 = = log 3 (2*(n new -1)/3 + 1) = = log 3 ((2*(n new -1)+3)/3) = = log 3 ((2*n new +1) - 1

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