A threaded binary tree node looks like following.

struct Node { struct Node *left, *right; int info; // True if left pointer points to predecessor // in Inorder Traversal bool lthread; // True if right pointer points to predecessor // in Inorder Traversal bool rthread; };

We have already discussed Insertion of Threaded Binary Search Tree

In deletion, first the key to be deleted is searched, and then there are differenct cases for deleting the Node in which key is found.

// Deletes a key from threaded BST with given root and // returns new root of BST. struct Node *delThreadedBST(struct Node* root, int dkey) { // Initialize parent as NULL and ptrent // Node as root. struct Node *par = NULL, *ptr = root; // Set true if key is found int found = 0; // Search key in BST : find Node and its // parent. while (ptr != NULL) { if (dkey == ptr->info) { found = 1; break; } par = ptr; if (dkey < ptr->info) { if (ptr->lthread == false) ptr = ptr -> left; else break; } else { if (ptr->rthread == false) ptr = ptr->right; else break; } } if (found == 0) printf("dkey not present in tree\n"); // Two Children else if (ptr->lthread == false && ptr->rthread == false) root = caseC(root, par, ptr); // Only Left Child else if (ptr->lthread == false) root = caseB(root, par, ptr); // Only Right Child else if (ptr->rthread == false) root = caseB(root, par, ptr); // No child else root = caseA(root, par, ptr); return root; }

**Case A: Leaf Node need to be deleted**

In BST, for deleting a leaf Node the left or right pointer of parent was set to NULL. Here instead of setting the pointer to NULL it is made a thread.

If the leaf Node is to be deleted is left child of its parent then after deletion, left pointer of parent should become a thread pointing to its predecessor of the parent Node after deletion.

par -> lthread = true; par -> left = ptr -> left;

If the leaf Node to be deleted is right child of its parent then after deletion, right pointer of parent should become a thread pointing to its successor. The Node which was inorder successor of the leaf Node before deletion will become the inorder successor of the parent Node after deletion.

// Here 'par' is pointer to parent Node and 'ptr' is // pointer to current Node. struct Node *caseA(struct Node *root, struct Node *par, struct Node *ptr) { // If Node to be deleted is root if (par == NULL) root = NULL; // If Node to be deleted is left // of its parent else if (ptr == par->left) { par->lthread = true; par->left = ptr->left; } else { par->rthread = true; par->right = ptr->right; } // Free memory and return new root free(ptr); return root; }

**Case B: Node to be deleted has only one child**

After deleting the Node as in a BST, the inorder successor and inorder predecessor of the Node are found out.

s = inSucc(ptr); p = inPred(ptr);

If Node to be deleted has left subtree, then after deletion right thread of its predecessor should point to its successor.

p->left = s;

Before deletion 15 is predecessor and 2 is successor of 16. After deletion of 16, the Node 20 becomes the successor of 15, so right thread of 15 will point to 20.

If Node to be deleted has right subtree, then after deletion left thread of its successor should point to its prredecessor.

s->left = p;

Before deletion of 25 is predecessor and 34 is successor of 30. After deletion of 30, the Node 25 becomes the predecessor of 34, so left thread of 34 will point to 25.

// Here 'par' is pointer to parent Node and 'ptr' is // pointer to current Node. struct Node *caseB(struct Node *root, struct Node *par, struct Node *ptr) { struct Node *child; // Initialize child Node to be deleted has // left child. if (ptr->lthread == false) child = ptr->left; // Node to be deleted has right child. else child = ptr->right; // Node to be deleted is root Node. if (par == NULL) root = child; // Node is left child of its parent. else if (ptr == par->left) par->left = child; else par->right = child; // Find successor and predecessor Node *s = inSucc(ptr); Node *p = inPred(ptr); // If ptr has left subtree. if (ptr->lthread == false) p->right = s; // If ptr has right subtree. else { if (ptr->rthread == false) s->left = p; } free(ptr); return root; }

**Case C: Node to be deleted has two children**

We find inorder successor of Node ptr (Node to be deleted) and then copy the information of this successor into Node ptr. After this inorder successor Node is deleted using either Case A or Case B.

// Here 'par' is pointer to parent Node and 'ptr' is // pointer to current Node. struct Node *caseC(struct Node *root, struct Node *par, struct Node *ptr) { // Find inorder successor and its parent. struct Node *parsucc = ptr; struct Node *succ = ptr -> right; // Find leftmost child of successor while (succ->left != NULL) { parsucc = succ; succ = succ -> left; } ptr->info = succ->info; if (succ->lthread == true && succ->rthread == true) root = caseA(root, parsucc, succ); else root = caseB(root, parsucc, succ); return root; }

Below is Complete C++ code:

// Complete C++ program to demonstrate deletion // in threaded BST #include<bits/stdc++.h> using namespace std; struct Node { struct Node *left, *right; int info; // True if left pointer points to predecessor // in Inorder Traversal bool lthread; // True if right pointer points to predecessor // in Inorder Traversal bool rthread; }; // Insert a Node in Binary Threaded Tree struct Node *insert(struct Node *root, int ikey) { // Searching for a Node with given value Node *ptr = root; Node *par = NULL; // Parent of key to be inserted while (ptr != NULL) { // If key already exists, return if (ikey == (ptr->info)) { printf("Duplicate Key !\n"); return root; } par = ptr; // Update parent pointer // Moving on left subtree. if (ikey < ptr->info) { if (ptr -> lthread == false) ptr = ptr -> left; else break; } // Moving on right subtree. else { if (ptr->rthread == false) ptr = ptr -> right; else break; } } // Create a new Node Node *tmp = new Node; tmp -> info = ikey; tmp -> lthread = true; tmp -> rthread = true; if (par == NULL) { root = tmp; tmp -> left = NULL; tmp -> right = NULL; } else if (ikey < (par -> info)) { tmp -> left = par -> left; tmp -> right = par; par -> lthread = false; par -> left = tmp; } else { tmp -> left = par; tmp -> right = par -> right; par -> rthread = false; par -> right = tmp; } return root; } // Returns inorder successor using left // and right children (Used in deletion) struct Node *inSucc(struct Node *ptr) { if (ptr->rthread == true) return ptr->right; ptr = ptr -> right; while (ptr->right) ptr = ptr->left; return ptr; } // Returns inorder successor using rthread // (Used in inorder) struct Node *inorderSuccessor(struct Node *ptr) { // If rthread is set, we can quickly find if (ptr -> rthread == true) return ptr->right; // Else return leftmost child of right subtree ptr = ptr -> right; while (ptr -> lthread == false) ptr = ptr -> left; return ptr; } // Printing the threaded tree void inorder(struct Node *root) { if (root == NULL) printf("Tree is empty"); // Reach leftmost Node struct Node *ptr = root; while (ptr -> lthread == false) ptr = ptr -> left; // One by one print successors while (ptr != NULL) { printf("%d ",ptr -> info); ptr = inorderSuccessor(ptr); } } struct Node *inPred(struct Node *ptr) { if (ptr->lthread == true) return ptr->right; ptr = ptr->left; while (ptr->rthread); ptr = ptr->right; return ptr; } // Here 'par' is pointer to parent Node and 'ptr' is // pointer to current Node. struct Node *caseA(struct Node *root, struct Node *par, struct Node *ptr) { // If Node to be deleted is root if (par == NULL) root = NULL; // If Node to be deleted is left // of its parent else if (ptr == par->left) { par->lthread = true; par->left = ptr->left; } else { par->rthread = true; par->right = ptr->right; } // Free memory and return new root free(ptr); return root; } // Here 'par' is pointer to parent Node and 'ptr' is // pointer to current Node. struct Node *caseB(struct Node *root, struct Node *par, struct Node *ptr) { struct Node *child; // Initialize child Node to be deleted has // left child. if (ptr->lthread == false) child = ptr->left; // Node to be deleted has right child. else child = ptr->right; // Node to be deleted is root Node. if (par == NULL) root = child; // Node is left child of its parent. else if (ptr == par->left) par->left = child; else par->right = child; // Find successor and predecessor Node *s = inSucc(ptr); Node *p = inPred(ptr); // If ptr has left subtree. if (ptr->lthread == false) p->right = s; // If ptr has right subtree. else { if (ptr->rthread == false) s->left = p; } free(ptr); return root; } // Here 'par' is pointer to parent Node and 'ptr' is // pointer to current Node. struct Node *caseC(struct Node *root, struct Node *par, struct Node *ptr) { // Find inorder successor and its parent. struct Node *parsucc = ptr; struct Node *succ = ptr -> right; // Find leftmost child of successor while (succ->left != NULL) { parsucc = succ; succ = succ -> left; } ptr->info = succ->info; if (succ->lthread == true && succ->rthread == true) root = caseA(root, parsucc, succ); else root = caseB(root, parsucc, succ); return root; } // Deletes a key from threaded BST with given root and // returns new root of BST. struct Node *delThreadedBST(struct Node* root, int dkey) { // Initialize parent as NULL and ptrent // Node as root. struct Node *par = NULL, *ptr = root; // Set true if key is found int found = 0; // Search key in BST : find Node and its // parent. while (ptr != NULL) { if (dkey == ptr->info) { found = 1; break; } par = ptr; if (dkey < ptr->info) { if (ptr->lthread == false) ptr = ptr -> left; else break; } else { if (ptr->rthread == false) ptr = ptr->right; else break; } } if (found == 0) printf("dkey not present in tree\n"); // Two Children else if (ptr->lthread == false && ptr->rthread == false) root = caseC(root, par, ptr); // Only Left Child else if (ptr->lthread == false) root = caseB(root, par, ptr); // Only Right Child else if (ptr->rthread == false) root = caseB(root, par, ptr); // No child else root = caseA(root, par, ptr); return root; } // Driver Program int main() { struct Node *root = NULL; root = insert(root, 20); root = insert(root, 10); root = insert(root, 30); root = insert(root, 5); root = insert(root, 16); root = insert(root, 14); root = insert(root, 17); root = insert(root, 13); root = delThreadedBST(root, 20); inorder(root); return 0; }

Output :

10 13 14 16 17 5 30

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