Descendants and ascendants in binary trees

A linear time and space algorithm for construction of a binary tree from the pre-order and post-order traversals is presented. The solution is not always unique. The number of solutions is calculated and an optimal time and space method to find all the solutions is shown.

This article introduces the basic concepts of binary trees, and then works through a series of practice problems with solution code in C/C++ and Java. Binary trees have an elegant recursive pointer structure, so they are a good way to learn recursive pointer algorithms.

The modeling of dynamical systems from a time series implemented by our DSA program introduces binary trees of height with all leaves on the same level, and the related subtrees of height L <= D. These are called epsilon-trees and epsilon-subtrees. The recursive and nonrecursive versions of the traversal algorithms for the trees with dynamically created nodes are discussed. The original nonrecursive algorithms that return the pointer to the next node in preorder, inorder and postorder traversals are presented. The space-time complexity analysis shows, and the execution time measurements confirm, that for these algorithms the recursive versions have approximately 10-25% better time constants. Still, the use of nonrecursive algorithms may be more appropriate in several occasions.

2012

Binary tree traversal refers to the process of visiting each node in a specified order. Given the inorder traversal of a binary tree, along with one of its preorder or postorder traversals, the original binary tree can be uniquely identified. Many recursive and non recursive method of construction of the tree from inorder and any of the postorder or preorder traversal have been proposed. In this paper one of the proposed algorithms has been examined. This algorithm computes the wrong tree for some input sequences. We show a particular situation in which the algorithm fails and a solution for this situation is proposed. The proposed a modified non-recursive algorithm for reconstructing a binary tree which generates the correct tree otherwise an error has been reported.

It is well-known that, given inorder traversal along with one of the preorder or postorder traversals of a binary tree, the tree can be determined uniquely. Several algorithms have been proposed to reconstruct a tree from its inorder and preorder traversals as well as inorder and postorder traversals. There is no study to focus on reconstructing a binary tree from both its preorder and postorder traversals. In this paper, we proved that given preorder and postorder traversals of a binary tree, the tree may not be identified uniquely, however, determining all the feasible solution(s) is possible. We present the PrePos algorithm, a novel algorithm to reconstruct all the possible binary tree(s) from its preorder and postorder traversals. PrePos algorithm not only finds the all the possible solutions, but also determines different types of the nodes in a binary tree; nodes with two children, nodes with one child, and node with no child. In the end, PrePos returns a matrix-based structure to represent all the binary tree solution(s). By this representation, the number of feasible solution can be counted in linear time.

A tree is a non-linear data structure for fast storing and retrieval of data in primary memory. It represents data in the form of hierarchical form. Data are stored in a tree i.e. called as a node, in which topmost node is called root and each node has one or more nodes lying on the left or right side of a tree. Except for root node each node has a parent node. The information can be extracted from a tree through various traversal algorithms. Tree traversal means visiting the nodes of a tree at once. In this paper, we are studying different algorithms for tree traversal

The Electronic Journal of Combinatorics, 2006

Simple families of increasing trees can be constructed from simply generated tree families, if one considers for every tree of size n all its increasing labellings, i. e. labellings of the nodes by distinct integers of the set f1;:::;ng in such a way that each sequence of labels along any branch starting at the root is increasing. Three such tree

Information Processing Letters, 1992

Gabrani, N. and P. Shankar, A note on the reconstruction of a binary tree from its traversals, Information Processing Letters 42 (1992) 117-119. We present a linear-time sequential algorithm for the construction of a binary tree, given its preorder and inorder traversals. The algorithm leads to an optimal O(log n) time parallel algorithm on the EREW PRAM model, where n is the number of nodes in the tree.

Information Processing Letters, 1977

International Journal of Computer Applications, 2012

Many reconstruction algorithms for binary tree have been discussed in this paper. A particular focus of this paper is on "A new Non-Recursive Algorithm for Reconstructing a Binary Tree from its Traversals". The computation time required for executing the reconstruction algorithm are O(N) and space complexity is O(NlogN) where N is the number of nodes in the binary tree. This algorithm works well for most of the input cases, but it has some drawbacks. There are some sequences of pre-order and in-order for which no legitimate tree can be constructed but the algorithm didn't take these cases into consideration and constructed a wrong tree for these cases. In this paper, we have proposed a solution to the problem in the previous algorithm and designed an algorithm which is the modified version of the previous algorithm for generating a correct binary tree. The new modified algorithm is implemented in C language and tested in GCC Compiler in Linux, for all types of input cases. The New modified algorithm works well for all types of input cases. We have calculated the best case time complexity of modified algorithm and show that a correct tree can be reported in O(N) time in best case and O(NlogN) space where N is the number of nodes in the tree. We have discussed some applications of the new modified algorithm in Huffman Coding, compiler design, text processing and searching algorithms.