Characteristic Inequalities for Binary Trees

In a binary tree T of N leaves, the left (right) level l i (r i) of leaf i is the number of left (right) edges in path from the root to that leaf. The level n i of leaf i is n i = l i +r i. Kraft-McMillan's characteristic inequality gives a necessary and suucient condition for a multiset of integers fn 1 ; n 2 ; :::; n N g to be the length ensemble of a binary tree T. Similarly, Yeung's characteristic inequality gives a necessary and suucient condition for a vector of integers (n 1 ; n 2 ; :::; n N) to be the length vector of a binary tree T. In this paper, we study characteristic inequalities for the path ensemble f N)) of a binary tree T. As an application of our results, we show a technique to construct U-R systems for the partial sums problem starting from weighted binary trees.