Trees:Threaded Binary Trees

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Threaded Binary Trees

Threads

Lemma 3.2 implies that a binary tree with n nodes has n + 1 null links. These null links can be replaced by pointers to nodes called threads. Threads are constructed using the following rules:

1. A null right child pointer in a node is replaced by a pointer to the inorder successor of p (i.e., the node that would be visited after p when traversing the tree inorder).

2. A null left child pointer in a node is replaced by a pointer to the inorder prede- cessor of p.

Figure 3.8 shows the binary tree of Figure 3.7 with threads drawn as broken lines. In order

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to distinguish between threads and normal pointers, two boolean fields LeftThread and RightThread are added to the node structure. If p->LeftThread is 1, then p->LeftChild contains a thread; otherwise it contains a pointer to the left child. Additionally, we assume that the tree contains a head node such that the original tree is the left subtree of the head node. The LeftChild pointer of node A and the RightChild pointer of node D point to the head node.

Inorder Traversal of a Threaded Binary Tree

Threads make it possible to perform an inorder traversal without using a stack. For any node p, if p’s right thread is 1, then its inorder successor is p->RightChild. Otherwise the inorder successor is obtained by following a path of left-child links from the right child of p until a node with left thread 1 is reached. Function Next below returns the inorder successor of currentNode (assuming that currentNode is not 0). It can be called repeatedly to traverse the entire tree in inorder in O(n) time. The code below assumes that the last node in the inorder traversal has a threaded right pointer to a dummy head node.

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Threads simplify the algorithms for preorder and postorder traversal. It is also possible to insert a node into a threaded tree in O(1) time [6].

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