Some results on Hu's conjecture concerning binary trees

Given n weights, w I, w2, . . . , w,, such that 0 G w 1 G w2 S. . . s w ,, we examine a property of permutation #, where 7~* = (w,, w,,, w2, w,,_r, . . .), concerning alphabetical binary trees. For each permutation ‘CT of these n weights, there is an optimal alphabetical binary tree corresponding to ‘TT, we denote it’s cost by V(m). There is also an optimal almost uniform alphabetical binary tree, corresponding to W, we denote it’s cost by VU(*). This paper asserts that V,(n*) > V,,(m) > V(T) for all 7~. This is a preliminary result concerning the conjecture of T.C. Hu. Hu’s conjecture is V(n*)a V(m) for all P.