On the inner structure of multidimensional simply generated trees

Abstract

AbstractThe well‐known concept of a “simply generated family of trees” is generalized to the multidimensional case. Such trees represent a symbiosis of trees and tries and they essentially arise in the construction of search trees for multidimensional keys. The set of nodes in a d‐dimensional simply generated tree can be partitioned into layers according to the nodes appearing in the ith dimension. We shall present a detailed average‐case analysis of these trees including the number of such trees of given size, the size of the ith layer, and the number of special nodes. The main result is that a multidimensional simply generated tree tends to a structure consisting of a collection of linear chains, on the average; in other words, for large;i ∈ [1: d], the ith layer consists only of one‐node trees.