  In this paper we study the redundancy of Huffman codes. In particular, we
consider sources for which the probability of one of the source symbols is
known. We prove a conjecture of Ye and Yeung regarding the upper bound on the
redundancy of such Huffman codes, which yields in a tight upper bound. We also
derive a tight lower bound for the redundancy under the same assumption.
  We further apply the method introduced in this paper to other related
problems. It is shown that several other previously known bounds with different
constraints follow immediately from our results.
