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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 >}}

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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 ->
```