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+---
+title: "Trees"
+weight: 1
+# bookFlatSection: false
+# bookToc: true
+# bookHidden: false
+bookCollapseSection: true
+# bookComments: false
+# bookSearchExclude: false
+---
+
+A tree, is a hierarchical way of organizing
+elements (often referred to as nodes) where each element has zero or more child elements. It is one
+of the most fundamental and widely used abstract data types (ADT). The structure consists of nodes
+connected by edges, with distinct properties:
+
+<!--more-->
+
+- **Root**: A special node at the top of a tree from which all other nodes descend. In some
+  implementations, there might not be a root if it's an empty tree.
+
+- **Parent and Child Nodes**: Each child node has one parent except for the root node, which doesn't
+  have any parents. There is exactly one edge between each pair of parent and its children (no shared
+  children).
+
+- **Leaf Nodes**: These are nodes that do not have any children. They represent the "end" points in
+  a tree structure.
+
+- **Edges/Links**: Connections between nodes, which can be directed or undirected. In binary trees
+  (a special kind of tree), an edge typically represents one possible path from a parent to its
+  child(ren).
+
+There are several types of trees that have specific properties and uses:
+
+- **Binary Trees**: Each node has at most two children, which can be named as the left child and
+  right child. Examples include Binary Search Trees (BST), AVL trees, Red-Black trees etc.
+
+- **Ternary Trees**: Each node may have up to three children. One common example is a Ternary Search
+  Tree used in text indexing.
+
+- **Balanced Trees**: These are binary trees that maintain their height as balanced with respect to
+  some metric (such as the number of nodes), like AVL and Red-Black trees, which help to ensure
+  operations on them run efficiently.
+
+- **B-trees and B+ Trees**: Non-binary tree structures used in databases and filesystems due to
+  their ability to handle large amounts of data with good performance for insertions, deletions, and
+  lookups.
+
+Trees are employed in various applications such as searching (e.g., binary search),
+sorting (in some cases using a heap structure which is a specific type of tree), managing
+hierarchical data, parsing expressions, routing protocols like Dijkstra's algorithm for finding the
+shortest path, and more.
+
+{{<section summary >}}
diff --git a/content/docs/dsa/trees/bst.md b/content/docs/dsa/trees/bst.md
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+---
+title: "Binary Search Tree"
+weight: 1
+# bookFlatSection: false
+# bookToc: true
+# bookHidden: false
+# bookCollapseSection: false
+# bookComments: false
+# bookSearchExclude: false
+---
+
+A Binary Search Tree (BST) is a type of data structure that organizes nodes in a hierarchical
+manner, where each node has at most two children: left and right. The key characteristic of a BST
+lies in the way it stores elements based on their values to maintain an ordered sequence that allows
+for efficient searching, insertion, and deletion operations.
+
+<!--more-->
+
+The fundamental properties of a binary search tree are as follows:
+
+1. **Node Structure**: Each node contains data (value), a reference to the left child node, and a
+   reference to the right child node. In addition, it may contain pointers for parent nodes in some
+   implementations but this is not mandatory.
+
+2. **Ordering Property**: For any given node in the BST, all values in its left subtree are less
+   than or equal to its own value, and all values in its right subtree are greater than its own value.
+   This property must hold for every single node, which is true recursively on each of its children as
+   well.
+
+3. **Efficiency**: Due to the ordering property, BSTs provide efficient time complexity for
+   operations like search (on average O(log n) in a balanced tree), insertion (O(log n)), and deletion
+   (O(log n)). However, these complexities can degrade to O(n) if the tree is not balanced.
+
+The efficiency of BSTs makes them useful for various applications that require sorted data storage
+with quick access times such as database indexing systems, sorting algorithms like heapsort and
+mergesort (when implemented using a binary heap), and many others in computer science.
+
+## Algorithm
+
+### Insertion
+
+1. **Start**: You are given the root of the BST and the integer value 'value' that needs to be
+   inserted into the tree.
+
+2. **Comparison with Root**: Begin by comparing the 'value' you wish to insert with the current root
+   node's value. If it is equal, skip the following steps as duplicates are not allowed in a BST (this
+   condition can vary based on specific implementation rules).
+
+3. **Decision for Insertion Location**:
+
+   - If the 'value' is less than the root node's value, move to the left child of the current node
+     and repeat step 2.
+   - If the 'value' is greater than the root node's value, move to the right child of the current
+     node and repeat step 2.
+
+4. **Find a Spot for New Node**: Continue this process of comparing the 'value' with each node it
+   encounters (left or right children) until an empty spot is found (a NULL pointer), which indicates
+   there is no child in that direction to insert before.
+
+5. **Insertion**: Once you reach a NULL position, create a new BSTNode object ('new_node') with the
+   'value' as its data and set it as either the left or right child of the last node visited (depending
+   on whether you moved left or right previously). This creates an insertion point in the tree.
+
+6. **End**: The algorithm ends here, and your BST now includes a new value at the correct position
+   according to its ordering property.
+
+### Deletion
+
+1. **Start**: You are given the root of the BST and the integer 'value' that needs to be removed.
+
+2. **Search for Target Node**: Traverse the tree starting from the root, comparing the target
+   'value' with each node’s value, moving left or right depending on whether it is less than or greater
+   than the current node's.
+
+3. **Case 1 - Leaf Node**: If the target node has no children (it is a leaf), simply remove it by
+   setting its parent's corresponding link to NULL.
+
+4. **Case 2 - Single Child**: If the target node has only one child, replace it with this child. For
+   example, if the left child exists, set the left child as root’s new left child and update the parent
+   reference of this child accordingly.
+
+5. **Case 3 - Both Children**: This is the most complex scenario since simply removing the node
+   might disrupt the BST properties. To maintain the tree structure after removal, you need to find
+   either the maximum value in the target's left subtree (to replace it as root of this subtree) or the
+   minimum value in the right subtree (which will take the place of the removed node). This replacement
+   ensures that the BST properties remain intact.
+
+6. **End**: The algorithm concludes, and you should now have a tree without the target 'value'.
+
+### Searching
+
+1. **Start**: You are provided with the root of the BST and the integer 'value'.
+
+2. **Initial Comparison**: Begin your search at the root node, comparing it against the target
+   value. If you reach a NULL pointer during this process (which implies that the tree is empty or
+   the element isn't present), stop further search as no match can be found in an empty tree.
+
+3. **Recursive Searching Process**: Depending on whether 'value' is less than, equal to, or
+   greater than the current node’s value, recursively move left if it's smaller, right if it's
+   larger, and return true (the element was found) if you encounter an exact match.
+
+4. **End of Search**: If at any point a comparison leads to an immediate equality check between
+   the target 'value' and the current node’s value, stop further search as the BST property
+   guarantees that this will be the only occurrence for duplicates (this step may vary based on
+   specific rules about duplicates in your implementation).
+
+5. **Outcome**: The algorithm concludes by either returning true if a match is found or false
+   otherwise, indicating whether 'value' exists within the tree or not.
+
+## Pseudocode
+
+```
+add(node, data) {
+   if (node == nullptr)
+      return create_node(data);
+   if (data < node -> data)
+      node -> left = insert_node(node -> left, data);
+   else if (data > node -> data)
+      node -> right = insert_node(node -> right, data);
+}
+
+remove_node(data) {
+   if (root == nullptr) return root;
+   if (data < root -> data)
+      root -> left = remove_node(root -> left, data);
+   else if (data > root -> data)
+      root -> right = remove_node(root -> right, data);
+   else {
+      // only child or no child
+      if (root -> left == nullptr) {
+         temp = root -> right;
+         root = nullptr;
+         delete root
+         return temp;
+      }
+      else if (root -> right == nullptr) {
+         temp = root -> left;
+         root = nullptr;
+         delete root;
+         return temp;
+      }
+      // two children
+      temp = smallest_node(root -> right);
+      root -> data = temp -> data;
+      root -> right = remove_node(root -> right, temp -> data);
+   }
+   return root;
+}
+
+search(root, data) {
+   if (root == null) return nullptr;
+   if (data == root -> data) return root -> data;
+   if (data < root -> data) return search(root -> left, data);
+   if (data > root -> data) return search(root -> right, data);
+   return root;
+}
+```
+## Code
+```cpp
+import <memory>;
+import <print>;
+
+struct Node;
+
+using node_ptr_t = std::shared_ptr<Node>;
+
+struct Node {
+  ssize_t data{};
+  node_ptr_t left{}, right{};
+
+  Node() = default;
+  Node(Node &&) = default;
+  explicit Node(ssize_t data, node_ptr_t left, node_ptr_t right)
+      : data(std::move(data)), left(left), right(right) {}
+  Node &operator=(Node &&) = default;
+  Node(const Node &) = delete;
+  Node &operator=(const Node &) = delete;
+};
+
+auto init_node(const ssize_t &data) -> node_ptr_t {
+  auto temp{std::make_shared<Node>()};
+  temp->data = data;
+  temp->left = nullptr;
+  temp->right = nullptr;
+  return temp;
+}
+
+auto travel_inorder(const node_ptr_t &root) -> void {
+  if (root != nullptr) {
+    travel_inorder(root->left);
+    std::print("{} -> ", root->data);
+    travel_inorder(root->right);
+  }
+}
+
+auto travel_preorder(const node_ptr_t &root) -> void {
+  if (root != nullptr) {
+    std::print("{} -> ", root->data);
+    travel_preorder(root->left);
+    travel_preorder(root->right);
+  }
+}
+
+auto travel_postorder(const node_ptr_t &root) -> void {
+  if (root != nullptr) {
+    travel_postorder(root->left);
+    travel_postorder(root->right);
+    std::print("{} -> ", root->data);
+  }
+}
+
+auto add_node(const node_ptr_t &node, const ssize_t &data) -> node_ptr_t {
+  if (node == nullptr)
+    return init_node(data);
+  if (data < node->data)
+    node->left = add_node(node->left, data);
+  else
+    node->right = add_node(node->right, data);
+  return node;
+}
+
+auto smallest_node(const node_ptr_t &given_node) -> node_ptr_t {
+  auto current_node{given_node};
+  // go to the leftmost node
+  while (current_node && current_node->left != nullptr)
+    current_node = current_node->left;
+  return current_node;
+}
+
+auto remove_node(node_ptr_t root, const ssize_t &data) -> node_ptr_t {
+  if (root == nullptr)
+    return root;
+  if (data < root->data)
+    root->left = remove_node(root->left, data);
+  else if (data > root->data)
+    root->right = remove_node(root->right, data);
+  else {
+    if (root->left == nullptr) {
+      auto temp{root->right};
+      return temp;
+    } else if (root->right == nullptr) {
+      auto temp{root->left};
+      return temp;
+    }
+
+    auto temp{smallest_node(root->right)};
+    root->data = temp->data;
+    root->right = remove_node(root->right, temp->data);
+  }
+  return root;
+}
+
+int main() {
+  node_ptr_t root{nullptr};
+  root = add_node(root, 8);
+  root = add_node(root, 5);
+  root = add_node(root, 2);
+  root = add_node(root, 6);
+  root = add_node(root, 7);
+  root = add_node(root, 1);
+  root = add_node(root, 25);
+  root = add_node(root, 54);
+  root = add_node(root, 4);
+  root = add_node(root, 11);
+  root = add_node(root, 9);
+  root = add_node(root, 3);
+
+  travel_inorder(root);
+  std::print("\n");
+  root = remove_node(root, 25);
+
+  travel_inorder(root);
+}
+```
+Here I have used smart pointers for automatic memory management.
+
+### Explanation
+1. **Headers and Type Aliases**
+```cpp
+import <memory>;
+import <print>;
+```
+These lines import the necessary standard library components: `memory` for `std::shared_ptr` and `print` for outputting text.
+
+```cpp
+struct Node;
+
+using node_ptr_t = std::shared_ptr<Node>;
+```
+This declares a forward declaration of the `Node` structure and a type alias `node_ptr_t` for a `std::shared_ptr<Node>`.
+
+2. **Node Structure**
+```cpp
+struct Node {
+  ssize_t data{};
+  node_ptr_t left{}, right{};
+
+  Node() = default;
+  Node(Node &&) = default;
+  explicit Node(ssize_t data, node_ptr_t left, node_ptr_t right)
+      : data(std::move(data)), left(left), right(right) {}
+  Node &operator=(Node &&) = default;
+  Node(const Node &) = delete;
+  Node &operator=(const Node &) = delete;
+};
+```
+The `Node` structure represents a node in the BST. Each node contains:
+
+- `data`: the value stored in the node.
+- `left` and `right`: pointers to the left and right children, respectively.
+
+The constructors and assignment operators are defined as follows:
+
+- Default constructor: `Node() = default`.
+- Move constructor and move assignment operator: `Node(Node &&) = default` and `Node &operator=(Node &&) = default`.
+- Parameterized constructor: initializes `data`, `left`, and `right`.
+- Copy constructor and copy assignment operator are deleted: `Node(const Node &) = delete` and `Node &operator=(const Node &) = delete` to prevent copying of nodes (only moving is allowed).
+
+3. **Initialize a Node**
+```cpp
+auto init_node(const ssize_t &data) -> node_ptr_t {
+  auto temp{std::make_shared<Node>()};
+  temp->data = data;
+  temp->left = nullptr;
+  temp->right = nullptr;
+  return temp;
+}
+```
+`init_node` creates and initializes a new node with the given data.
+
+4. **Tree Traversal Functions**
+```cpp
+auto travel_inorder(const node_ptr_t &root) -> void {
+  if (root != nullptr) {
+    travel_inorder(root->left);
+    std::print("{} -> ", root->data);
+    travel_inorder(root->right);
+  }
+}
+
+auto travel_preorder(const node_ptr_t &root) -> void {
+  if (root != nullptr) {
+    std::print("{} -> ", root->data);
+    travel_preorder(root->left);
+    travel_preorder(root->right);
+  }
+}
+
+auto travel_postorder(const node_ptr_t &root) -> void {
+  if (root != nullptr) {
+    travel_postorder(root->left);
+    travel_postorder(root->right);
+    std::print("{} -> ", root->data);
+  }
+}
+```
+These functions implement in-order, pre-order, and post-order traversal of the BST, respectively, and print the nodes' data during traversal.
+
+4. **Add Node to Tree**
+```cpp
+auto add_node(const node_ptr_t &node, const ssize_t &data) -> node_ptr_t {
+  if (node == nullptr)
+    return init_node(data);
+  if (data < node->data)
+    node->left = add_node(node->left, data);
+  else
+    node->right = add_node(node->right, data);
+  return node;
+}
+```
+`add_node` recursively adds a new node with the given data to the BST, maintaining the BST property.
+
+5. **Find the Smallest Node**
+```cpp
+auto smallest_node(const node_ptr_t &given_node) -> node_ptr_t {
+  auto current_node{given_node};
+  while (current_node && current_node->left != nullptr)
+    current_node = current_node->left;
+  return current_node;
+}
+```
+`smallest_node` finds and returns the node with the smallest value in the subtree rooted at `given_node`.
+
+6. **Remove Node from Tree**
+```cpp
+auto remove_node(node_ptr_t root, const ssize_t &data) -> node_ptr_t {
+  if (root == nullptr)
+    return root;
+  if (data < root->data)
+    root->left = remove_node(root->left, data);
+  else if (data > root->data)
+    root->right = remove_node(root->right, data);
+  else {
+    if (root->left == nullptr) {
+      auto temp{root->right};
+      return temp;
+    } else if (root->right == nullptr) {
+      auto temp{root->left};
+      return temp;
+    }
+
+    auto temp{smallest_node(root->right)};
+    root->data = temp->data;
+    root->right = remove_node(root->right, temp->data);
+  }
+  return root;
+}
+```
+`remove_node` removes a node with the specified data from the BST. It handles three cases:
+
+- Node with only one child or no child.
+- Node with two children: finds the in-order successor (smallest node in the right subtree), replaces the node's data with the successor's data, and then deletes the successor.
+
+7. **`main()` Function**
+```cpp
+int main() {
+  node_ptr_t root{nullptr};
+  root = add_node(root, 8);
+  root = add_node(root, 5);
+  root = add_node(root, 2);
+  root = add_node(root, 6);
+  root = add_node(root, 7);
+  root = add_node(root, 1);
+  root = add_node(root, 25);
+  root = add_node(root, 54);
+  root = add_node(root, 4);
+  root = add_node(root, 11);
+  root = add_node(root, 9);
+  root = add_node(root, 3);
+
+  travel_inorder(root);
+  std::print("\n");
+  root = remove_node(root, 25);
+
+  travel_inorder(root);
+}
+```
+The `main` function demonstrates creating a BST, adding nodes to it, performing an in-order traversal, removing a node, and performing another in-order traversal.
+
+## Output
+```console
+1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 11 -> 25 -> 54 -> 
+1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 -> 8 -> 9 -> 11 -> 54 -> 
+```
\ No newline at end of file