### Insertion in an m-Way search tree:

The insertion in an m-Way search tree is similar to binary trees but there should be no more than **m-1** elements in a node. If the node is full then a child node will be created to insert the further elements.

Let us see the example given below to insert an element in an m-Way search tree.

**Example: **

- To insert a new element into an m-Way search tree we proceed in the same way as one would in order to search for the element
- To insert 6 into the 5-Way search tree shown in the figure, we proceed to search for 6 and find that we fall off the tree at the node [7, 12] with the first child node showing a null pointer
- Since the node has only two keys and a 5-Way search tree can accommodate up to 4 keys in a node, 6 is inserted into the node like [6, 7, 12]
- But to insert 146, the node [148, 151, 172, 186] is already full, hence we open a new child node and insert 146 into it. Both these insertions have been illustrated below

`// Inserts a value in the m-Way tree ` `struct` `node* insert(` `int` `val, `
` ` `struct` `node* root) `
`{ ` ` ` `int` `i; `
` ` `struct` `node *c, *n; `
` ` `int` `flag; `
` ` ` ` `// Function setval() is called which `
` ` `// returns a value 0 if the new value `
` ` `// is inserted in the tree, otherwise `
` ` `// it returns a value 1 `
` ` `flag = setval(val, root, &i, &c); `
` ` ` ` `if` `(flag) { `
` ` `n = (` `struct` `node*)` `malloc` `(` `sizeof` `(` `struct` `node)); `
` ` `n->count = 1; `
` ` `n->value[1] = i; `
` ` `n->child[0] = root; `
` ` `n->child[1] = c; `
` ` `return` `n; `
` ` `} `
` ` `return` `root; `
`} ` |

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**insert(): **

- The function
**insert()**receives two parameters-the address of the new node and the value that is inserted - This function calls a function
**setval()**which returns a value**0**if the new value is inserted in the tree, otherwise it returns a value**1** - If it returns
**1**then memory is allocated for new node, the variable**count**is assigned a value**1**and the new value is inserted in the node - Then the addresses of the child nodes are stored in
**child**pointers and finally the address of the node is returned

`// Sets the value in the node ` `int` `setval(` `int` `val, `
` ` `struct` `node* n, `
` ` `int` `* p, `
` ` `struct` `node** c) `
`{ ` ` ` `int` `k; `
` ` ` ` `// if node is null `
` ` `if` `(n == NULL) { `
` ` `*p = val; `
` ` `*c = NULL; `
` ` `return` `1; `
` ` `} `
` ` `else` `{ `
` ` ` ` `// Checks whether the value to be `
` ` `// inserted is present or not `
` ` `if` `(searchnode(val, n, &k)) `
` ` `printf` `(` `"Key value already exists\n"` `); `
` ` ` ` `// The if-else condition checks whether `
` ` `// the number of nodes is greater or less `
` ` `// than the maximum number. If it is less `
` ` `// then it inserts the new value in the `
` ` `// same level node, otherwise, it splits the `
` ` `// node and then inserts the value `
` ` `if` `(setval(val, n->child[k], p, c)) { `
` ` ` ` `// if the count is less than the max `
` ` `if` `(n->count < MAX) { `
` ` `fillnode(*p, *c, n, k); `
` ` `return` `0; `
` ` `} `
` ` `else` `{ `
` ` ` ` `// Insert by splitting `
` ` `split(*p, *c, n, k, p, c); `
` ` `return` `1; `
` ` `} `
` ` `} `
` ` `return` `0; `
` ` `} `
`} ` |

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**setval(): **

- The function
**setval()**receives four parameters- The first argument is the value that is to be inserted
- The second argument is the address of the node
- The third argument is an integer pointer that points to a local flag variable defined in the function
**insert()** - The last argument is a pointer to pointer to the child node that will be set in a function called from this function

- The function
**setval()**returns a flag value that indicates whether the value is inserted or not - If the node is empty then this function calls a function
**searchnode()**that checks whether the value already exists in the tree - If the value already exists then a suitable message is displayed
- Then a recursive call is made to the function
**setval()**for the child of the node - If this time the function returns a value 1 it means the value is not inserted
- Then a condition is checked whether the node is full or not
- If the node is not full then a function
**fillnode()**is called that fills the value in the node hence at this point a value 0 is returned - If the node is full then a function
**split()**called that splits the existing node. At this point, a value 1 is returned to add the current value to the new node

`// Adjusts the value of the node ` `void` `fillnode(` `int` `val, `
` ` `struct` `node* c, `
` ` `struct` `node* n, `
` ` `int` `k) `
`{ ` ` ` `int` `i; `
` ` ` ` `// Shifting the node by one position `
` ` `for` `(i = n->count; i > k; i--) { `
` ` `n->value[i + 1] = n->value[i]; `
` ` `n->child[i + 1] = n->child[i]; `
` ` `} `
` ` `n->value[k + 1] = val; `
` ` `n->child[k + 1] = c; `
` ` `n->count++; `
`} ` |

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**fillnode(): **

- The function
**fillnode()**receives four parameters- The first is the value that is to be inserted
- The second is the address of the child node of the new value that is to be inserted
- The third is the address of the node in which the new value is to be inserted
- The last parameter is the position of the node where the new value is to be inserted

`// Splits the node ` `void` `split(` `int` `val, `
` ` `struct` `node* c, `
` ` `struct` `node* n, `
` ` `int` `k, ` `int` `* y, `
` ` `struct` `node** newnode) `
`{ ` ` ` `int` `i, mid; `
` ` `if` `(k <= MIN) `
` ` `mid = MIN; `
` ` `else`
` ` `mid = MIN + 1; `
` ` ` ` `// Allocating the memory for a new node `
` ` `*newnode = (` `struct` `node*) `
`malloc` `(` `sizeof` `(` `struct` `node)); `
` ` ` ` `for` `(i = mid + 1; i <= MAX; i++) { `
` ` `(*newnode)->value[i - mid] = n->value[i]; `
` ` `(*newnode)->child[i - mid] = n->child[i]; `
` ` `} `
` ` ` ` `(*newnode)->count = MAX - mid; `
` ` `n->count = mid; `
` ` ` ` `// it checks whether the new value `
` ` `// that is to be inserted is inserted `
` ` `// at a position less than or equal `
` ` `// to minimum values required in a node `
` ` `if` `(k <= MIN) `
` ` `fillnode(val, c, n, k); `
` ` `else`
` ` `fillnode(val, c, *newnode, k - mid); `
` ` ` ` `*y = n->value[n->count]; `
` ` `(*newnode)->child[0] = n->child[n->count]; `
` ` `n->count--; `
`} ` |

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**split(): **

- The function
**split()**receives six parameters- The first four parameters are exactly the same as in the case of function
**fillnode()** - The fifth parameter is a pointer to variable that holds the value from where the node is split
- The last parameter is a pointer to pointer of the new node created at the time of split

- The first four parameters are exactly the same as in the case of function

**MIN**

**MIN**

### Deletion in an m-Way search tree:

Let **K** be a key to be deleted from the m-Way search tree. To delete the key we proceed as one would to search for the key. Let the node accommodating the key be as illustrated below.

**Approach: **

There are several cases for deletion

- If
**(A**then delete K_{i}= A_{j}= NULL) - If
**(A**then choose the largest of the key elements_{i}!= NULL, A_{j}= NULL)**K’**in the child node pointed to by**A**, delete the key_{i}**K’**and replace**K**by**K’** - Obviously deletion of
**K’**may call for subsequent replacements and therefore deletions in a similar manner, to enable the key**K’**move up the tree - If
**(A**then choose the smallest of the key element_{i}= NULL, A_{j}!= NULL)**K”**from the sub-tree pointed to by**A**, delete_{j}**K”**and replace**K**by**K”** - Again deletion of
**K”**may trigger subsequent replacements and deletions to enable**K”**move up the tree - If
**(A**then choose either the largest of the key elements_{i}!=NULL, A_{j}!= NULL)**K’**in the sub-tree pointed to by**A**, or the smallest of the key elements_{i}**K”**from the sub-tree pointed to by**A**to replace K_{j} - As mentioned before, to move
**K’**or**K”**up the tree it may call for subsequent replacements and deletions

**Example: **

- To delete 151, we search for 151 and observe that in the leaf node [148, 151, 172, 186] where it is present, both its left sub-tree pointer and right sub-tree pointer are such that
**(A**_{i}= A_{j}= NULL) - We therefore simply delete 151 and the node becomes [148, 172, 186]. Deletion of 92 also follows a similar process
- To delete 262, we find its left and right sub-tree pointers A
_{i</sub and Aj respectively, are such that (Ai = Aj = NULL)} - Hence we choose the smallest element 272 from the child node [272, 286, 350], delete 272 and replace 262 with 272. Note that, to delete 272 the deletion procedure needs to be observed again
- To delete 12, we find the node [7, 12] accommodates 12 and the key satisfies
**(A**_{i}!= NULL, A_{j}= NULL) - Hence we choose the largest of the keys from the node pointed to by Ai, viz., 10 and replace 12 with 10. This deletion is illustrated below

This deletion is illustrated below

`// Deletes value from the node ` `struct` `node* del(` `int` `val, `
` ` `struct` `node* root) `
`{ ` ` ` `struct` `node* temp; `
` ` `if` `(!delhelp(val, root)) { `
` ` `printf` `(` `"\n"` `); `
` ` `printf` `(` `"value %d not found.\n"` `, val); `
` ` `} `
` ` `else` `{ `
` ` `if` `(root->count == 0) { `
` ` `temp = root; `
` ` `root = root->child[0]; `
` ` `free` `(temp); `
` ` `} `
` ` `} `
` ` `return` `root; `
`} ` |

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**del(): **

- The function
**del()**receives two parameters. First is the value that is to be deleted second is the address of the root node - This function calls another helper function
**delhelp()**which returns value 0 if the deletion of the value is unsuccessful, 1 otherwise - Otherwise, a condition is checked whether the
**count**is 0 - If it is, then it indicates that the node from which the value is deleted is the last value
- Hence, the first child of the node is itself made the node and the original node is deleted. Finally, the address of the new root node is returned

`// Helper function for del() ` `int` `delhelp(` `int` `val, `
` ` `struct` `node* root) `
`{ ` ` ` `int` `i; `
` ` `int` `flag; `
` ` `if` `(root == NULL) `
` ` `return` `0; `
` ` `else` `{ `
` ` ` ` `// Again searches for the node `
` ` `flag = searchnode(val, `
` ` `root, `
` ` `&i); `
` ` ` ` `// if flag is true `
` ` `if` `(flag) { `
` ` `if` `(root->child[i - 1]) { `
` ` `copysucc(root, i); `
` ` `// delhelp() is called recursively `
` ` `flag = delhelp(root->value[i], `
` ` `root->child[i]) `
` ` `if` `(!flag) `
` ` `{ `
` ` `printf` `(` `"\n"` `); `
` ` `printf` `(` `"value %d not found.\n"` `, val); `
` ` `} `
` ` `} `
` ` `else`
` ` `clear(root, i); `
` ` `} `
` ` `else` `{ `
` ` `// Recursion `
` ` `flag = delhelp(val, root->child[i]); `
` ` `} `
` ` ` ` `if` `(root->child[i] != NULL) { `
` ` `if` `(root->child[i]->count < MIN) `
` ` `restore(root, i); `
` ` `} `
` ` `return` `flag; `
` ` `} `
`} ` |

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**delhelp(): **

- The function
**delhelp()**receives two parameters. First is the value to be deleted and the second is the address of the node from which it is to be deleted - Initially it is checked whether the node is NULL
- If it is, then a value 0 is returned
- Otherwise, a call to function
**searchnode()**is made - If the value is found then another condition is checked to see whether there is any child to the value that is to be deleted
- If so, the function
**copysucc()**is called which copies the successor of the value to be deleted and then a recursive call is made to the function**delhelp()**for the value that is to be deleted and its child - If the child is empty then a call to function
**clear()**is made which deletes the value - If the
**searchnode()**function fails then a recursive call is made to function**delhelp()**by passing the address of the child - If the child of the node is not empty, then a function
**restore()**is called to merge the child with its siblings - Finally the value of the
**flag**is returned which is set as a returned value of the function**searchnode()**

`// Removes the value from the ` `// node and adjusts the values ` `void` `clear(` `struct` `node* m, ` `int` `k) `
`{ ` ` ` `int` `i; `
` ` `for` `(i = k + 1; i <= m->count; i++) { `
` ` `m->value[i - 1] = m->value[i]; `
` ` `m->child[i - 1] = m->child[i]; `
` ` `} `
` ` `m->count--; `
`} ` |

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**clear(): **

- The function
**clear()**receives two parameters. First is the address of the node from which the value is to be deleted and second is the position of the value that is to be deleted - This function simply shifts the values one place to the left from the position where the value is to be deleted is present

`// Copies the successor of the ` `// value that is to be deleted ` `void` `copysucc(` `struct` `node* m, ` `int` `i) `
`{ ` ` ` `struct` `node* temp; `
` ` `temp = p->child[i]; `
` ` `while` `(temp->child[0]) `
` ` `temp = temp->child[0]; `
` ` `p->value[i] = temp->value[i]; `
`} ` |

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**copysucc()**

- The function
**copysucc()**receives two parameters. First is the address of the node where the successor is to be copied and second is the position of the value that is to be overwritten with its successor

`// Adjusts the node ` `void` `restore(` `struct` `node* m, ` `int` `i) `
`{ ` ` ` `if` `(i == 0) { `
` ` `if` `(m->child[1]->count > MIN) `
` ` `leftshift(m, 1); `
` ` `else`
` ` `merge(m, 1); `
` ` `} `
` ` `else` `{ `
` ` `if` `(i == m->count) { `
` ` `if` `(m->child[i - 1]->count > MIN) `
` ` `rightshift(m, i); `
` ` `else`
` ` `merge(m, i); `
` ` `} `
` ` `else` `{ `
` ` `if` `(m->child[i - 1]->count > MIN) `
` ` `rightshift(m, i); `
` ` `else` `{ `
` ` `if` `(m->child[i + 1]->count > MIN) `
` ` `leftshift(m, i + 1); `
` ` `else`
` ` `merge(m, i); `
` ` `} `
` ` `} `
` ` `} `
`} ` |

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**restore(): **

- The function
**restore()**receives two parameters. First is the node that is to be restored and second is the position of the value from where the values are restored - If the second parameter is 0, then another condition is checked to find out whether the values present at the first child are more than the required minimum number of values
- If so, then a function
**leftshift()**is called by passing the address of the node and a value 1 signifying that the value of this node is to be shifted from the first value - If the condition is not satisfied then a funcition
**merge()**is called for merging the two children of the node

`// Adjusts the values and children ` `// while shifting the value from ` `// parent to right child ` `void` `rightshift(` `struct` `node* m, ` `int` `k) `
`{ ` ` ` `int` `i; `
` ` `struct` `node* temp; `
` ` ` ` `temp = m->child[k]; `
` ` ` ` `// Copying the nodes `
` ` `for` `(i = temp->count; i > 0; i--) { `
` ` `temp->value[i + 1] = temp->value[i]; `
` ` `temp->child[i + 1] = temp->child[i]; `
` ` `} `
` ` `temp->child[1] = temp->child[0]; `
` ` `temp->count++; `
` ` `temp->value[1] = m->value[k]; `
` ` ` ` `temp = m->child[k - 1]; `
` ` `m->value[k] = temp->value[temp->count]; `
` ` `m->child[k]->child[0] `
` ` `= temp->child[temp->count]; `
` ` `temp->count--; `
`} ` ` ` `// Adjusts the values and children ` `// while shifting the value from ` `// parent to left child ` `void` `leftshift(` `struct` `node* m, ` `int` `k) `
`{ ` ` ` `int` `i; `
` ` `struct` `node* temp; `
` ` ` ` `temp = m->child[k - 1]; `
` ` `temp->count++; `
` ` `temp->value[temp->count] = m->value[k]; `
` ` `temp->child[temp->count] `
` ` `= m->child[k]->child[0]; `
` ` ` ` `temp = m->child[k]; `
` ` `m->value[k] = temp->value[1]; `
` ` `temp->child[0] = temp->child[1]; `
` ` `temp->count--; `
` ` ` ` `for` `(i = 1; i <= temp->count; i++) { `
` ` `temp->value[i] = temp->value[i + 1]; `
` ` `temp->child[i] = temp->child[i + 1]; `
` ` `} `
`} ` |

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**rightshift() and leftshift()**

- The function
**rightshift()**receives two parameters - First is the address of the node from where the value is shifted to its child and second is the position
**k**of the value that is to be shifted - The function
**leftshift()**are exactly same as that of function**rightshift()** - The function
**merge()**receives two parameters. First is the address of the node from which the value is to copied to the child and second is the position of the value

`// Merges two nodes ` `void` `merge(` `struct` `node* m, ` `int` `k) `
`{ ` ` ` `int` `i; `
` ` `struct` `node *temp1, *temp2; `
` ` ` ` `temp1 = m->child[k]; `
` ` `temp2 = m->child[k - 1]; `
` ` `temp2->count++; `
` ` `temp2->value[temp2->count] = m->value[k]; `
` ` `temp2->child[temp2->count] = m->child[0]; `
` ` ` ` `for` `(i = 0; i <= temp1->count; i++) { `
` ` `temp2->count++; `
` ` `temp2->value[temp2->count] = temp1->value[i]; `
` ` `temp2->child[temp2->count] = temp1->child[i]; `
` ` `} `
` ` `for` `(i = k; i < m->count; i++) { `
` ` `m->value[i] = m->value[i + 1]; `
` ` `m->child[i] = m->child[i + 1]; `
` ` `} `
` ` `m->count--; `
` ` `free` `(temp1); `
`} ` |

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- The function
**merge()**receives two parameters - First is the address of the node from which the value is to be copied to the child and second is the position of the value
- In this function, two temporary variables temp1 and temp2 are defined to hold the addresses of the two children of the value that is to be copied
- Initially, the value of the node is copied to its child. Then the first child of the node is made the respective child of the node where the value is copied
- Then two for loops are executed, out of which first copies all the values and children of one child to the other child
- The second loop shifts the value and child of the node from where the value is copied
- Then the count of the node from where the node is copied is decremented. Finally, the memory occupied by the second node is released by calling
**free()**

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## Recommended Posts:

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- Ternary Search Tree (Deletion)
- Complexity of different operations in Binary tree, Binary Search Tree and AVL tree
- AVL Tree | Set 2 (Deletion)
- Deletion in a Binary Tree
- Van Emde Boas Tree | Set 4 | Deletion
- Deletion of a given node K in a Binary Tree using Level Order Traversal
- ScapeGoat Tree | Set 1 (Introduction and Insertion)
- Proto Van Emde Boas Tree | Set 3 | Insertion and isMember Query
- Insertion in n-ary tree in given order and Level order traversal
- Van Emde Boas Tree | Set 2 | Insertion, Find, Minimum and Maximum Queries
- AVL Tree | Set 1 (Insertion)
- Threaded Binary Tree | Insertion
- Insertion in a Binary Tree in level order
- Optimal sequence for AVL tree insertion (without any rotations)
- Insertion in a B+ tree
- Skip List | Set 3 (Searching and Deletion)
- Fibonacci Heap - Deletion, Extract min and Decrease key
- Difference between Binary Tree and Binary Search Tree
- Proto Van Emde Boas Trees | Set 4 | Deletion

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