2c03-review - 00068

# 2c03-review - 00068 - 13[20 Suppose T is a Huffman tree and...

This preview shows page 1. Sign up to view the full content.

11 13.[20] Suppose T is a Huffman tree, and that the leaf for symbol a has greater depth than the leaf for symbol b . Prove that the probability of symbol b is no less than that of a . It can be proven in many ways by induction. The simplest seems to be the following. Let a 1 , … , a n , be a sequence of characters. Without loss of generality we may assume that prob(a i ) prob(a j ) if i<j, so a 1 , a 2 are characters with the smallest probabilities. First we prove the following lemma: “a 1 , a 2 are siblings nodes whose depth is at least as big as any other leaf in the tree”. Suppose that a 1 and a 2 are not the deepest nodes in the tree. In this case, the Huffman tree must either look as the below one, or be symmetrical to this. v u a 1 a 2 x For this situation to occur, the parent of a 1 and a 2 , labeled v, must have greater combined probability than the node labeled x. Otherwise we would have selected v in place of node x as the child of u. However, this is impossible since a 1 and a 2 are the symbols with the smallest probabilities. Hence our lemma is true.
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