vault backup: 2025-03-16 18:59:42

Data Structures/AI Notes/Trees.md

@@ -9,29 +9,17 @@ Linear lists, which include arrays and linked lists, are ideal for serially orde
 ### Examples of Hierarchical Data in Trees
 
 - Corporate Structure:
-    
-
 - In a company, positions such as president, vice presidents, managers, etc., can be represented in a tree format.
-    
-
 - Object-Oriented Programming:
-    
-
 - In Java, classes form a hierarchy where the Object class is at the top, followed by subclasses.
-    
 
 ## Tree Terminologies
 
 Understanding tree terminologies is crucial for grasping more complex operations and implementations.
 
 - **Depth:** The number of edges from the root node to a specific node. For example, if node G is three levels down from the root, its depth is 3.
-    
-
 - **Height:** The height of a tree is defined as the number of levels it possesses. A tree's maximum depth helps determine its overall height.
-    
-
 - **Degree of a Node:** This refers to the number of child nodes a specific node has. A tree where each node has a maximum of two children is termed a **binary tree**.
-    
 
 ### Binary Trees
 
@@ -42,13 +30,8 @@ A binary tree is defined such that each node has at most two children, known tra
 When analyzing binary trees, certain properties can be established:
 
 - A binary tree of height h must have at least h nodes.
-    
-
 - In a full binary tree, each node has either two children or none. It contains the maximum number of nodes defined as 2^h - 1.
-    
-
 - The number of nodes and tree height relationship is succinctly expressed with the inequalities: h \leq n \leq 2^h -1.
-    
 
 ## Binary Operations and Traversals
 
@@ -59,47 +42,31 @@ Performing operations on binary trees is essential to manage and manipulate data
 Traversal techniques allow for processing each node in a specified order:
 
 - **Pre-order Traversal:** Process the root, then traverse the left subtree, followed by the right subtree.
-    
-
 - **In-order Traversal:** Visit the left subtree first, then process the root, followed by the right subtree.
-    
-
 - **Post-order Traversal:** Traverse the left subtree, right subtree, and then process the root.
-    
-
 - **Level-order Traversal:** Process nodes level by level from the root down to the leaves.
-    
 
 ### Arithmetic Expressions as Trees
 
-Binary trees are particularly useful for representing arithmetic expressions. An _expression tree_ is structured such that:
+Binary trees are particularly useful for representing arithmetic expressions. An *expression tree* is structured such that:
 
 - Leaves represent operands.
-    
-
 - Non-leaf nodes denote operators.
-    
-
 - Subtrees represent sub-expressions.
-    
 
 ## Conversion Between Expression Formats
 
 Moving between infix, prefix, and postfix expressions requires constructing expression trees first.
 
 1. Infix expression (a + b) can be converted to a postfix expression (a b +) by building the corresponding expression tree and performing a post-order traversal.
-    
 
 ## Constructing Binary Trees from Traversals
 
 Creating a binary tree from given traversals (like in-order and pre-order) involves recognizing patterns in the data:
 
 - From in-order and pre-order traversals, one identifies the root and splits left and right subtrees accordingly.
-    
-
 - The process can illustrate various scenarios and result in multiple valid trees for the same traversal combinations.
-    
 
 ## Conclusion
 
-Tree data structures are indispensable in computer science, providing efficient methods for data organization, retrieval, and management. Understanding their properties and operations enables clearer problem-solving strategies and effective programming techniques.
+Tree data structures are indispensable in computer science, providing efficient methods for data organization, retrieval, and management. Understanding their properties and operations enables clearer problem-solving strategies and effective programming techniques.