Fibonacci connection between Huffman codes and Wythoff array

A non-decreasing sequence of positive integer weights P = {p1, . . . , p2, pn} is called k-ordered if an intermediate sequence of weights produced by Huffman algorithm for initial sequence P on i-th step satisfies the following conditions: p (i) 2 = p (i) 3 , i = 0, k; p (i) 2 < p (i) 3 , i = k + 1, n − 3. Let T be a binary tree of size n and M = M(T ) be a set of such sequences of positive integer weights that the tree T is the Huffman tree of P (|P | = n). A sequence Pmin of n positive integer weights is called a minimizing sequence of the binary tree T in class M(Pmin ∈ M) if Pmin produces the minimal Huffman cost of the tree T over all sequences from M , i.e., E(T, Pmin) ≤ E(T, P ) ∀P ∈ M . Fibonacci related connection between minimizing k-ordered sequences of the maximum height Huffman tree and the Wythoff array [Sloane, A035513] has been proved. Let Mn,k (k = 0, n − 3) denote the set of all k-ordered sequences of size n for which the Huffman tree has maximum height. Let F (i) denote i-th Fibonacci number. Theorem: A minimizing k-ordered sequence of the maximum height Huffman tree in class Mn,k (k = 0, n − 3) is Pminn,k = {p1, p2, . . . , pn}, where p1 = 1, p2 = F (1), . . . , pk+2 = F (k + 1), pk+3 = F (k + 2) = wF (k+2),0, pk+4 = wF (k+2),1, pk+5 = wF (k+2),2, . . . , pn = wF (k+2),n−k−3; wi,j is (i, j)th element of the Wythoff array. The cost of Huffman trees for those sequences has been computed. Several examples of minimizing ordered sequences for Huffman codes are shown. 1 Main Conceptions and Terminology 1.1 Binary Trees A (strictly) binary tree is an oriented ordered tree where each nonleaf node has exactly two children (siblings). A binary tree is called elongated if at least one of any two sibling nodes is a leaf. An elongated binary tree of size n has maximum height among all binary trees of size n. An elongated binary tree is called left− sided if the right node in each pair of sibling nodes is a leaf. A binary tree is called labeled if a certain positive integer (weight) is set in correspondence with each leaf. 1. Main Conceptions and Terminology Size of a tree is the total number of leaves of this tree. Definition. Let T be a binary tree with positive weights P = {p1, p2, . . . , pn} at its leaf nodes. The weighted external path length of T is