Binary Tree

Understanding Binary Tree

To kickstart this journey into the realm of computer science, let's first delve into what Binary Tree is - a fundamental data structure used for computer science applications like image processing algorithms and efficient searches.

Essential Properties of Binary Tree

• Root: The topmost node in the tree.
• Child: Any node, excluding the root, connected upward by an edge.
• Parent: For any 'child' node, the connected node above it, nearer to the root.
• Leaf: A node with no children.
• Subtree: Any node can be viewed as the root, together with children, of a subtree.
• Visiting: Executing some form of operation i.e., print, compute a function at the node.
• Traversing: Visiting the nodes in some order.
• Levels: levels are numbered from the root down. The root is at level one.

Binary Tree Structure Explained

• Full Binary Tree: This type includes a tree where every node has either zero or two children.
• Complete Binary Tree: In this type, all levels are completely filled, except possibly the last level which is filled from left to right.

Diving Into Binary Tree Search

Binary Tree Search is a classic computer science algorithm that is used to locate a specific value within a binary tree. The algorithm operates on the principle that for each node, all elements in its left subtree are less than the node, and all elements in the right subtree are greater than the node. Binary Tree Search operates by efficiently halving the search space at each step, redefining it to either the left or right subtree, based on the comparison.

Implementing Search in Binary Tree

Binary Tree Search in Python

Here's a Python code snippet showing the implementation of Binary Tree Search:

class Node:    
def __init__(self, key):        
self.left = None 
self.right = None 
self.val = key 
def search(root, key):    
# Base Cases: root is null or key is present at root    
if root is None or root.val == key:        
return root    
# Key is greater than root's key 
if root.val < key:        
return search(root.right, key)     
# Key is smaller than root's key 
return search(root.left, key)

In the snippet above, a Node class is created, and the BST search operation is implemented as a function which takes two parameters – 'root' and 'key'. The function tests whether the currently explored node's key matches the searched key. In case of a match, it returns the node.

If the searched key is greater, the function is recursively called on the right subtree, and otherwise, on the left subtree. If the search space is exhausted (i.e., an empty subtree is reached), 'None' is returned signifying the key has not been found in the tree.

Exploring Binary Tree Examples

• Search Systems: One of the most widespread applications of Binary Trees is in search systems. For instance, in an online dictionary, Binary Tree Search allows a user to find a word much faster as compared to linear searching.
• Decision Trees: Decision trees, a popular tool in decision analysis, are also a type of binary tree. Each node represents a decision, and the tree’s structure allows for a comprehensive review of the result of each decision path.
• Sorting Algorithms: Binary Trees are also used in sorting algorithms like heapsort that are more efficient than bubble, insertion, and selection sort.

Binary Tree Examples in Computer Science

• File System Hierarchy: The file system in computer systems is structured in the form of a Binary Tree. Each folder (directory) and file is a node of the tree structure. The Binary Tree's hierarchical property suits naturally for this task given the inherent hierarchy of file systems.
• Expression Parsing: Expression parsing, a crucial aspect of compiler design, deploys Binary Tree. The expression is represented as a Binary Tree structure, allowing the compiler to easily evaluate the expression.
• Network Data Routing: Network routers use a binary space partitioning tree to speed up data access and foster efficient routing in the network backbone.

A Guide to Inverting Binary Tree

What is an Inverted Binary Tree?

How to Invert a Binary Tree

1. Swap the left child and the right child of the root node.
2. Recursively invert the subtrees rooted at the just swapped left and right children.

Inverting Binary Tree in Python

class Node:    
def __init__(self, key):        
self.left = None 
self.right = None 
self.val = key 
def invert(node):    
if node is None:        
return
# Swap left and right child nodes 
temp = node.left    
node.left = node.right    
node.right = temp     
# Recursive call on subtrees 
invert(node.left)    
invert(node.right)

Mastering Balanced Binary Tree

A Balanced Binary Tree is a particular type of Binary Tree that is deeply ingrained in computer science due to its unique and useful properties. The primary attribute that defines a Balanced Binary Tree is the fact that the height of the left and right subtrees of every node differs by at most 1.

This ideal balancement condition ensures optimal performance in operations like insertion, deletion, and search, and therefore, Balanced Binary Trees are used extensively in various data-lookup structures like AVL trees, red-black trees and B-trees.

Balancing a Binary Tree: An Outline

• Left Rotation: Performed when the right subtree of a node is heavier (higher).
• Right Rotation: Performed when the left subtree of a node is heavier.
• Left-Right Rotation: A combination of left rotation followed by right rotation. Performed when the left subtree is heavier and its right child contains the imbalance.
• Right-Left Rotation: A combination of right rotation followed by left rotation. Performed when the right subtree is heavier and its left child contains the imbalance.

Balancing Binary Trees in Python

Here’s a short Python code snippet showcasing this:

class Node:    
def __init__(self, key, height=1):        
self.left = None 
self.right = None 
self.val = key        
self.height = height # height of node added
def insert(node, key):
#...insertion code... 
# Update height of node 
node.height = 1 + max(getHeight(node.left), getHeight(node.right))    
# Get balance to check if tree became unbalanced    
balance = getBalance(node)    
# Perform rotations (4 cases) 
# ...rotation code...

In the above-mentioned code, a Node class has been defined equipped with an additional attribute – height, which stores the height of the node.

After every insert operation, the height of the node is updated, and the balance is checked. If the balance exceeds 1, appropriate rotations are performed based on whether the imbalance is in the left, right, left-right, or right-left subtrees. This iterative process thereby ensures that the Binary Tree always remains balanced.

Implementing the Binary Tree in Python

class Node:    
def __init__(self, data):        
self.left = None        
self.right = None 
self.data = dat

def insert(root, data):
if root is None:
return Node(data)
else:
if data < root.data:
root.left = insert(root.left, data)
else: 
root.right = insert(root.right, data)
return roo

def inOrderTraversal(root):
if root:        
inOrderTraversal(root.left)        
print(root.data)        
inOrderTraversal(root.right)

Python Binary Tree: Simple Steps to Get Started

1. Create a Node class: Each node of the Binary Tree will be an instance of this class, containing a 'data' attribute, and two nodes 'left' and 'right' representing the left and right children respectively.
2. Design the 'insert' function: This function takes the root of a Binary Tree and some data. It inserts the data into the Binary Tree rooted at 'root', obeying the Binary Search Tree property, and returns the root.
3. Create a 'traversal' function: This function prints the nodes' data in various orders. The function takes a root, traverses the Binary tree rooted at 'root' in in-order (left --> root --> right), pre-order (root --> left --> right), or post-order (left --> right --> root), and prints the data.

Handling Binary Tree Data using Python

Once you have understood the basic implementation of the Binary Tree, it opens gates to a whole lot of operations and manipulations of Binary Trees. Be it balancing, rotating, or Mirroring the Binary Tree; you can do it all. To maintain the Binary Tree as an efficient, ordered data structure, we need to ensure that the tree remains balanced, primarily when we add or remove nodes. Python provides several built-in functions to simplify these tasks.

We discussed binary tree balancing and rotation in detail in previous sections. The key is to ensure that the maximum difference in the height of the left and the right subtrees of any node is one. Observing this rule will keep the tree balanced and optimise the time complexity of searching, insertion, and deletion operations to \(O(log(n))\). To mirror or invert a Binary Tree, again Python's straightforwardness comes into play.

Here's a neat way to invert a Binary Tree:

def invert(root): 
if root: 
root.left, root.right = root.right, root.left 
invert(root.left) 
invert(root.right) 
return root

In this Python function, at every node encountered, its left and right children are swapped. The function then recursively inverts the left and right subtrees. Couldn't get any simpler, right?