On the expected number of nodes at level k in 0-balanced trees

Abstract

An ordered tree with height n and m leaves is called 0-balanced if all leaves have the same level. We compute the average number of nodes (with specified degree) appearing at a given level in a 0-balanced ordered tree as well as in a 0-balanced t-ary ordered tree.

With respect to the former class we shall show that the average rate of increase of nodes amounts to gr := (m - 1)n ⁻¹ passing from one level to the next one. The same fact holds for nodes with a degree one at a large level. The average number of nodes with a degree two and that one with a degree greater than two tends to ρ and zero for large levels, respectively.

The class of 0-balanced t-ary trees corresponds to the set of all code trees associated with all n-block codes with m code words over a given alphabet with cardinality t. In that case, we shall show that all nodes with maximal degree t are concentrated at levels smaller than log_t(m), and all nodes with degree one appear at levels greater than log, (m), on the average. This result implies that the first log, (m) positions in the m code words appearing in an n-block code over a given alphabet with cardinality t are sufficient to separate these words, on the average, provided that all those codes are equally likely.