{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# SLN_2 - Problem 1 A heap of size n has at most n/2h 1 nodes...

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

Problem 1 A heap of size n has at most d n/ 2 h +1 e nodes with height h . Proof : Property 1: Let S h be the set of nodes of height h , subtrees rooted at nodes in S h are disjoint. In other words, we cannot have two nodes of height h with one being an ancestor of the other. Property 2 All subtrees are complete binary trees except for one subtree. Now we derive the bounds of n by N h given these two properties. Let N h be the number of nodes of height h . Since N h - 1 of these subtrees are full, each subtree of them contains exactly 2 h +1 - 1 nodes. One of the height h subtrees may be not full, but contain at least 1 node at its lower level and has at least 2 h nodes. The remaining nodes have height strictly more than h . To connect all subtrees rooted at node in S h , there must be exactly N h - 1 such nodes. The total of nodes is at least ( N h - 1)(2 h +1 - 1)+2 h + N h - 1 while at most N h 2 h +1 - 1, So ( N h - 1)(2 h +1 - 1) + 2 h + ( N h - 1) n N h (2 h +1 - 1) + N h - 1 (1) ⇒ - 2 h n - N h 2 h +1 ≤ - 1 (2) The fraction part of n/ 2 h +1

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 ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern