Recursion Depth Analysis for Specail Tree Traversal Algorithms

In this paper we are concerned with the analysis of special recursive algorithms for traversing the nodes of a planted plane tree (ordered tree ; planar tree) . Some by now classical results in this area are due to KNUTH [7], DE BRUIJN, KNUTH and RICE [1], FLAJOLET [2], FLAJOLET and ODLYZKO [3], FLAJOLET, RAOULT and VUILLEMIN [4], KEMP [6] and others and are summarized in the next few lines : The most important .tree structure in Computer Science are the binary trees . The inorder traversal (KNUTH [7]) is the following recursive principle : Traverse the left subtree Visit the root Traverse the right subtree . The most straightforward implementation uses an auxiliary stack to keep necessary nodes of the tree . . The analysis of the expected time of the visit procedure is clearly linear in the size of the input tree . To evaluate recursion depth means to determine the average stack height as a function of the size of the tree . The recursion depth or height h of the binary tree is recursively determined as follows : If the family B of binary trees is given by the symbolic equation