docs: Complete AVL tree README

Author: 4ndrelim

docs/assets/images/AvlTree.png

@@ -55,7 +55,7 @@ public int height(T key) {
     }
 
     /**
-     * Update height of node in avl tree during rebalancing.
+     * Update height of node in avl tree for re-balancing.
      *
      * @param n node whose height is to be updated
      */
@@ -372,6 +372,10 @@ private T successor(Node<T> node) {
         return null;
     }
 
+
+    // ---------------------------------------------- NOTE ------------------------------------------------------------
+    // METHODS BELOW ARE NOT NECESSARY; JUST FOR VISUALISATION PURPOSES
+
     /**
      * prints in order traversal of the entire tree.
      */
@@ -390,13 +394,9 @@ private void printInorder(Node<T> node) {
         if (node == null) {
             return;
         }
-        if (node.getLeft() != null) {
-            printInorder(node.getLeft());
-        }
+        printInorder(node.getLeft());
         System.out.print(node + " ");
-        if (node.getRight() != null) {
-            printInorder(node.getRight());
-        }
+        printInorder(node.getRight());
     }
 
     /**
@@ -408,7 +408,6 @@ public void printPreorder() {
         System.out.println();
     }
 
-
     /**
      * Prints out pre-order traversal of tree rooted at node
      *
@@ -419,12 +418,8 @@ private void printPreorder(Node<T> node) {
             return;
         }
         System.out.print(node + " ");
-        if (node.getLeft() != null) {
-            printPreorder(node.getLeft());
-        }
-        if (node.getRight() != null) {
-            printPreorder(node.getRight());
-        }
+        printPreorder(node.getLeft());
+        printPreorder(node.getRight());
     }
 
     /**
@@ -442,12 +437,11 @@ public void printPostorder() {
      * @param node node which the tree is rooted at
      */
     private void printPostorder(Node<T> node) {
-        if (node.getLeft() != null) {
-            printPostorder(node.getLeft());
-        }
-        if (node.getRight() != null) {
-            printPostorder(node.getRight());
+        if (node == null) {
+            return;
         }
+        printPostorder(node.getLeft());
+        printPostorder(node.getRight());
         System.out.print(node + " ");
     }
 

docs/assets/images/BalancedProof.png

@@ -15,10 +15,9 @@ public class Node<T extends Comparable<T>> {
     private Node<T> parent;
     private int height;
     /*
-     * Can insert more properties here.
-     * If key is not unique, introduce a value property
-     * so when nodes are being compared, a distinction
-     * can be made
+     * Can insert more properties here for augmentation
+     * e.g. If key is not unique, introduce a value property as a tie-breaker
+     * or weight property for order statistics
      */
 
     public Node(T key) {

docs/assets/images/TreeRotation.png

@@ -0,0 +1,90 @@
+# AVL Trees
+
+## Background
+Is the fastest way to search for data to store them in an array, sort them and perform binary search? No. This will
+incur minimally O(nlogn) sorting cost, and O(n) cost per insertion to maintain sorted order.
+
+We have seen binary search trees (BSTs), which always maintains data in sorted order. This allows us to avoid the
+overhead of sorting before we search. However, we also learnt that unbalanced BSTs can be incredibly inefficient for
+insertion, deletion and search operations, which are O(h) in time complexity (in the case of degenerate trees,
+operations can go up to O(n)).
+
+Here we discuss a type of self-balancing BST, known as the AVL tree, that avoids the worst case O(n) performance 
+across the operations by ensuring careful updating of the tree's structure whenever there is a change 
+(e.g. insert or delete).
+
+### Definition of Balanced Trees
+Balanced trees are a special subset of trees with **height in the order of log(n)**, where n is the number of nodes. 
+This choice is not an arbitrary one. It can be mathematically shown that a binary tree of n nodes has height of at least
+log(n) (in the case of a complete binary tree). So, it makes intuitive sense to give trees whose heights are roughly
+ in the order of log(n) the desirable 'balanced' label.
+
+<div align="center">
+    <img src="../../../../../docs/assets/images/BalancedProof.png" width="40%">
+    <br>
+    Credits: CS2040s Lecture 9
+</div>
+
+### Height-Balanced Property of AVL Trees
+There are several ways to achieve a balanced tree. Red-black tree, B-Trees, Scapegoat and AVL trees ensure balance 
+differently. Each of them relies on some underlying 'good' property to maintain balance - a careful segmenting of nodes 
+in the case of RB-trees and enforcing a depth constraint for B-Trees. Go check them out in the other folders! <br>
+What is important is that this **'good' property holds even after every change** (insert/update/delete).
+
+The 'good' property in AVL Trees is the **height-balanced** property. Height-balanced on a node is defined as  
+**difference in height between the left and right child node being not more than 1**. <br>
+We say the tree is height-balanced if every node in the tree is height-balanced. Be careful not to conflate 
+the concept of "balanced tree" and "height-balanced" property. They are not the same; the latter is used to achieve the
+former.
+
+<details>
+<summary> <b>Ponder..</b> </summary>
+Consider any two nodes (need not have the same immediate parent node) in the tree. Is the difference in height 
+between the two nodes <= 1 too?
+</details>
+
+It can be mathematically shown that a **height-balanced tree with n nodes, has at most height <= 2log(n)**. Therefore, 
+following the definition of a balanced tree, AVL trees are balanced.
+
+<div align="center">
+    <img src="../../../../../docs/assets/images/AvlTree.png" width="40%">
+    <br>
+    Credits: CS2040s Lecture 9
+</div>
+
+## Complexity Analysis
+**Search, Insertion, Deletion, Predecessor & Successor queries Time**: O(height) = O(logn)
+
+**Space**: O(n) <br>
+where n is the number of elements (whatever the structure, it must store at least n nodes)
+
+## Operations
+Minimally, an implementation of AVL tree must support the standard **insert**, **delete**, and **search** operations. 
+**Update** can be simulated by searching for the old key, deleting it, and then inserting a node with the new key. 
+
+Naturally, with insertions and deletions, the structure of the tree will change, and it may not satisfy the 
+"height-balance" property of the AVL tree. Without this property, we may lose our O(log(n)) run-time guarantee. 
+Hence, we need some re-balancing operations. To do so, tree rotation operations are introduced. Below is one example.
+
+<div align="center">
+    <img src="../../../../../docs/assets/images/TreeRotation.png" width="40%">
+    <br>
+    Credits: CS2040s Lecture 10
+</div>
+
+Prof Seth explains it best! Go re-visit his slides (Lecture 10) for the operations :P <br>
+Here is a [link](https://www.youtube.com/watch?v=dS02_IuZPes&list=PLgpwqdiEMkHA0pU_uspC6N88RwMpt9rC8&index=9) 
+for prof's lecture on trees. <br>
+_We may add a summary in the near future._
+
+## Application
+While AVL trees offer excellent lookup, insertion, and deletion times due to their strict balancing, 
+the overhead of maintaining this balance can make them less preferred for applications 
+where insertions and deletions are significantly more frequent than lookups. As a result, AVL trees often find itself
+over-shadowed in practical use by other counterparts like RB-trees, 
+which boast a relatively simple implementation and lower overhead, or B-trees which are ideal for optimizing disk 
+accesses in databases.
+
+That said, AVL tree is conceptually simple and often used as the base template for further augmentation to tackle 
+niche problems. Orthogonal Range Searching and Interval Trees can be implemented with some minor augmentation to 
+an existing AVL tree.