The Number of Descendants in Simply Generated Random Trees

The aim of this note is to generalize some recent results for binary trees by Panholzer and Prodinger [15] to a larger class of rooted trees. The number of descendants of a node j is the number of nodes in the subtree rooted at j, and the number of ascendants is the number of nodes between j and the root. Recently, Panholzer and Prodinger [15] studied the behavior of these parameters in binary trees during various traversal algorithms. The case of binary search trees was treated by Mart́ınez, Panholzer and Prodinger [14]. In this paper we will study the number of descendants in simply generated trees (defined below). The number of ascendants is already treated in [1] and [10]. Let us start with a description of the traversal algorithms we will investigate. In the binary case there are basically three traversal algorithms. All of them are recursive algorithms treating the left subtree before the right subtree. They differ with respect to the visit of the root: first (preorder), middle (inorder), and last (postorder). We will study the number of descendants in simply generated trees during preorder and postorder traversal. Since the outdegree of any node in a simply generated tree need not be equal to zero or two, inorder traversal cannot be well defined for that class of trees. Let us recall the definition of simply generated trees. Let A be a class of plane rooted trees and define for T ∈ A the size |T | by the number of nodes of T . Furthermore there is assigned a weight ω(T ) to each T ∈ A. Let an denote the quantity an = ∑