techbase-python/lessons/lesson_39.html at main · techbaseng/techbase-python · GitHub

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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8"><meta name="viewport" content="width=device-width,initial-scale=1.0">
<title>Lesson 39: Trees, Binary Trees, Binary Search Trees, AVL Trees, and Graphs in Python — Techbase Python</title>
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.se ul{list-style:none;display:flex;flex-direction:column;gap:.45rem}
.se li{font-size:.95rem;opacity:.93;display:flex;align-items:center;gap:.5rem}
.ln-btn{display:inline-flex;align-items:center;gap:.4rem;background:var(--sf);
  color:var(--t2);font-family:var(--D);font-weight:600;font-size:.88rem;
  padding:.7rem 1.3rem;border-radius:8px;text-decoration:none;
  border:1.5px solid var(--br);transition:border-color .2s,color .2s}
.ln-btn:hover{border-color:var(--acc);color:var(--acc)}
.ln-btn.primary{background:var(--acc);color:#fff;border-color:var(--acc)}
.ln-btn.primary:hover{opacity:.9}
.ln{display:flex;align-items:center;justify-content:space-between;
  padding-top:1.5rem;border-top:1px solid rgba(255,255,255,.25);
  margin-top:1.5rem;flex-wrap:wrap;gap:.75rem}

/* UNLOCK BUTTON */
.ub{display:inline-flex;align-items:center;gap:.5rem;background:var(--acc);
  color:#fff;font-family:var(--D);font-weight:700;font-size:.95rem;
  padding:.8rem 1.7rem;border-radius:10px;border:none;cursor:pointer;
  transition:transform .2s,box-shadow .2s;box-shadow:0 4px 14px var(--acs);margin-top:1.3rem}
.ub:hover{transform:translateY(-2px);box-shadow:0 8px 22px var(--acs)}
.ub:active{transform:scale(.97)}

/* TOAST */
#toast{position:fixed;bottom:95px;right:1.5rem;z-index:999;background:#1e293b;color:#fff;
  border-radius:10px;padding:.8rem 1.3rem;font-family:var(--D);font-size:.9rem;
  font-weight:600;box-shadow:0 8px 26px rgba(0,0,0,.25);
  transform:translateY(20px);opacity:0;transition:all .3s;pointer-events:none}
#toast.show{transform:translateY(0);opacity:1}

/* GO TO TOP */
#go-top{position:fixed;bottom:1.5rem;right:1.5rem;z-index:300;
  width:48px;height:48px;border-radius:50%;background:var(--acc);color:#fff;
  border:none;cursor:pointer;font-size:1.2rem;box-shadow:0 4px 16px var(--acs);
  display:flex;align-items:center;justify-content:center;
  opacity:0;transform:translateY(10px);transition:opacity .3s,transform .3s,box-shadow .2s;
  pointer-events:none}
#go-top.visible{opacity:1;transform:translateY(0);pointer-events:auto}
#go-top:hover{box-shadow:0 8px 26px var(--acs);transform:translateY(-2px)}

/* CONFETTI */
#cc{position:fixed;top:0;left:0;width:100%;height:100%;pointer-events:none;z-index:998;display:none}

/* FOOTER */
.lf{display:flex;align-items:center;justify-content:center;gap:1rem;
  padding:1.6rem 1.5rem;border-top:1px solid var(--br);background:var(--sf)}
.lf img{height:32px;width:auto;opacity:.7}
.lf-text{font-family:var(--M);font-size:.75rem;color:var(--mu)}
</style>
</head>
<body>
<nav class="l-nav">
  <a href="../index.html" class="nav-logo-wrap">
    <img src="data:image/png;base64,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" 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" alt="Techbase Consultant Services">
    </div>
    <div class="l-hero-text">
      <div class="badge">Python · Lesson 39</div>
      <h1>Trees, Binary Trees, Binary Search Trees, AVL Trees, and Graphs in Python</h1>
      <div class="l-hero-sub">10 phases &nbsp;·&nbsp; Build: Project Brief</div>
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  <div style="max-width:880px;margin:0 auto;padding:0 1.5rem"><div class="wb"><h2>👋 Welcome to Lesson 39</h2><div style="font-size:1rem;line-height:1.85;color:var(--t2)"><p class="v2-p">Welcome to one of the most exciting topics in Data Structures and Algorithms! In this lesson, you will learn about <strong>Trees</strong> and <strong>Graphs</strong> · two of the most powerful and widely-used data structures in computing.</p>
<p class="v2-p">You have already learned about <strong>linear data structures</strong>: Arrays, Linked Lists, Stacks, and Queues. In those structures, elements follow one after another in a straight line.</p>
<p class="v2-p">But what happens when your data is not a simple list? What if it has <strong>branches</strong>, <strong>hierarchies</strong>, or <strong>relationships</strong> that cannot be captured in a straight line?</p>
<p class="v2-p">That is exactly what <strong>Trees</strong> and <strong>Graphs</strong> solve. They let you represent data that branches out · like a family tree, a folder system on your computer, a city's road map, or even the internet itself.</p>
<p class="v2-p">By the end of this lesson, you will be able to:</p>
<ul class="v2-ul"><li>Understand what a Tree is and why it is useful</li><li>Build and traverse Binary Trees</li><li>Implement a Binary Search Tree (BST) with insert, search, and delete</li><li>Understand how AVL Trees keep themselves balanced automatically</li><li>Understand Graphs and how they are stored in memory</li></ul>
<p class="v2-p">Let's start from the very beginning · no prior knowledge of trees or graphs is needed!</p>
<hr class="v2-hr"></div><div class="wm"><span>📚 10 phases</span><span>🏗️ Project Brief</span><span>🐍 GitHub Repo</span></div></div></div>
</section>

<main style="max-width:880px;margin:0 auto;padding:2rem 1.5rem 5rem">
<div class="phase" id="phase1"><div class="ph"><div class="pn">Phase 1 of 10</div><div class="pt">Lesson Introduction</div><div class="pc" id="chk1"></div></div><div class="pb2"><p class="v2-p">Welcome to one of the most exciting topics in Data Structures and Algorithms! In this lesson, you will learn about <strong>Trees</strong> and <strong>Graphs</strong> · two of the most powerful and widely-used data structures in computing.</p>
<p class="v2-p">You have already learned about <strong>linear data structures</strong>: Arrays, Linked Lists, Stacks, and Queues. In those structures, elements follow one after another in a straight line.</p>
<p class="v2-p">But what happens when your data is not a simple list? What if it has <strong>branches</strong>, <strong>hierarchies</strong>, or <strong>relationships</strong> that cannot be captured in a straight line?</p>
<p class="v2-p">That is exactly what <strong>Trees</strong> and <strong>Graphs</strong> solve. They let you represent data that branches out · like a family tree, a folder system on your computer, a city's road map, or even the internet itself.</p>
<p class="v2-p">By the end of this lesson, you will be able to:</p>
<ul class="v2-ul"><li>Understand what a Tree is and why it is useful</li><li>Build and traverse Binary Trees</li><li>Implement a Binary Search Tree (BST) with insert, search, and delete</li><li>Understand how AVL Trees keep themselves balanced automatically</li><li>Understand Graphs and how they are stored in memory</li></ul>
<p class="v2-p">Let's start from the very beginning · no prior knowledge of trees or graphs is needed!</p>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>Lesson Introduction</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(1)">Start Lesson ✓</button></div></div>
<div class="phase locked" id="phase2"><div class="ph"><div class="pn">Phase 2 of 10</div><div class="pt">Prerequisite Concepts</div><div class="pc" id="chk2"></div></div><div class="pb2"><p class="v2-p">Before diving in, let us quickly review the concepts you already know that will help here.</p>
<p class="v2-p"><strong>Classes and Objects:</strong> Trees and Graphs are built using Python classes. Each "node" in a tree is an object.</p>
<p class="v2-p"><strong>Recursion:</strong> Many tree operations use recursion · a function that calls itself. Think of it like a set of Russian nesting dolls: you open one, and inside is a smaller version of the same thing.</p>
<p class="v2-p"><strong>Linked Lists:</strong> A Binary Tree works similarly to a Linked List, except instead of one "next" pointer, each node has a "left" and a "right" pointer.</p>
<p class="v2-p">If any of those sound unfamiliar, don't worry · we will explain everything as we go!</p>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>Prerequisite Concepts</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(2)">Phase Complete — Unlock Next ✓</button></div></div>
<div class="phase locked" id="phase3"><div class="ph"><div class="pn">Phase 3 of 10</div><div class="pt">Part 1: What Is a Tree?</div><div class="pc" id="chk3"></div></div><div class="pb2"><h3 class="v2-h3">Understanding Trees · The Big Picture</h3>
<p class="v2-p">Imagine your company's organisational chart. At the very top is the CEO. Below the CEO are vice-presidents. Below each vice-president are managers. Below each manager are employees. This is a <strong>hierarchy</strong> · a structure that branches downward.</p>
<p class="v2-p">That is exactly what a <strong>Tree data structure</strong> looks like!</p>
<p class="v2-p">A <strong>Tree</strong> is a data structure made of <strong>nodes</strong> (boxes that hold values) connected by <strong>edges</strong> (lines between boxes). It is called a "tree" because it looks like an upside-down tree: the root is at the top, and the branches spread downward.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>        R          &lt;-- This is the ROOT (the top node)
       / \
      A   B        &lt;-- A and B are children of R
     / \ / \
    C  D E  F      &lt;-- Leaf nodes (no children)
              \
               G   &lt;-- G is a child of F</code></pre></div></div>
<h3 class="v2-h3">Key Vocabulary · Explained Simply</h3>
<div class="v2-table-wrap"><table class="v2-table"><thead><tr><th>Word</th><th>Simple Meaning</th></tr></thead><tbody><tr><td><strong>Node</strong></td><td>A box that holds a value</td></tr><tr><td><strong>Root</strong></td><td>The very top node (the starting point)</td></tr><tr><td><strong>Child</strong></td><td>A node connected below another node</td></tr><tr><td><strong>Parent</strong></td><td>The node directly above a child node</td></tr><tr><td><strong>Leaf</strong></td><td>A node with no children (at the bottom)</td></tr><tr><td><strong>Edge</strong></td><td>The line connecting two nodes</td></tr><tr><td><strong>Height</strong></td><td>The number of edges from root to the farthest leaf</td></tr><tr><td><strong>Subtree</strong></td><td>Any node plus all nodes below it</td></tr></tbody></table></div>
<h3 class="v2-h3">Why Not Just Use Arrays or Linked Lists?</h3>
<p class="v2-p">Great question! Here is the honest comparison:</p>
<ul class="v2-ul"><li><strong>Arrays</strong> are fast at accessing a specific element directly (e.g., element 700 in 1000 elements). But inserting or deleting requires shifting other elements in memory · slow!</li><li><strong>Linked Lists</strong> are fast at inserting and deleting. But finding an element means going through the entire list one by one · slow!</li><li><strong>Trees</strong> give you the BEST of both worlds. Accessing, inserting, and deleting can all be done <strong>fast</strong>, without shifting elements in memory.</li></ul>
<h3 class="v2-h3">Real-World Uses of Trees</h3>
<p class="v2-p">Trees appear everywhere in computing and real life:</p>
<ul class="v2-ul"><li><strong>File systems:</strong> Your computer's folders and files form a tree (the root is <code>/</code> or <code>C:\</code>)</li><li><strong>Databases:</strong> Used for quick data lookup</li><li><strong>Routing tables:</strong> Networking uses trees to decide how to send data</li><li><strong>Sorting and searching:</strong> Trees make these operations super fast</li><li><strong>AI game engines:</strong> Chess and other game AIs build decision trees</li><li><strong>Priority queues:</strong> Often built using a special type of tree called a heap</li></ul>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>: What Is a Tree?</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(3)">Phase Complete — Unlock Next ✓</button></div></div>
<div class="phase locked" id="phase4"><div class="ph"><div class="pn">Phase 4 of 10</div><div class="pt">Part 2: Binary Trees</div><div class="pc" id="chk4"></div></div><div class="pb2"><h3 class="v2-h3">What Makes a Tree "Binary"?</h3>
<p class="v2-p">A <strong>Binary Tree</strong> is a tree where <strong>every node has at most two children</strong> · a <strong>left child</strong> and a <strong>right child</strong>.</p>
<p class="v2-p">The word "binary" means "two". So binary tree = each node can branch into at most two directions.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>        R
       / \
      A   B
     / \ / \
    C  D E  F
              \
               G</code></pre></div></div>
<p class="v2-p">Why is this useful? By limiting each node to at most two children, operations like searching, inserting, and deleting become much easier to program and much faster to run.</p>
<h3 class="v2-h3">Building a Binary Tree in Python</h3>
<p class="v2-p">Let us now write code to create a Binary Tree! We will use a Python <strong>class</strong> for the node.</p>
<h4 class="v2-h4">Step 1: Create the TreeNode class</h4>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>class TreeNode:
    def __init__(self, data):
        self.data = data      # The value stored in this node
        self.left = None      # Pointer to the left child (starts as nothing)
        self.right = None     # Pointer to the right child (starts as nothing)</code></pre></div></div>
<p class="v2-p"><strong>Line by line explanation:</strong></p>
<ul class="v2-ul"><li><code>class TreeNode:</code> · We are defining a blueprint for a tree node</li><li><code>def __init__(self, data):</code> · This runs automatically when we create a new node</li><li><code>self.data = data</code> · Store the value passed in (like 'R' or 42)</li><li><code>self.left = None</code> · No left child yet (None means "nothing")</li><li><code>self.right = None</code> · No right child yet</li></ul>
<h4 class="v2-h4">Step 2: Create all the nodes and link them</h4>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>root = TreeNode(&#x27;R&#x27;)
nodeA = TreeNode(&#x27;A&#x27;)
nodeB = TreeNode(&#x27;B&#x27;)
nodeC = TreeNode(&#x27;C&#x27;)
nodeD = TreeNode(&#x27;D&#x27;)
nodeE = TreeNode(&#x27;E&#x27;)
nodeF = TreeNode(&#x27;F&#x27;)
nodeG = TreeNode(&#x27;G&#x27;)

# Connect the nodes
root.left = nodeA     # R&#x27;s left child is A
root.right = nodeB    # R&#x27;s right child is B

nodeA.left = nodeC    # A&#x27;s left child is C
nodeA.right = nodeD   # A&#x27;s right child is D

nodeB.left = nodeE    # B&#x27;s left child is E
nodeB.right = nodeF   # B&#x27;s right child is F

nodeF.left = nodeG    # F&#x27;s left child is G

# Test: Print a specific node&#x27;s value
print(&quot;root.right.left.data:&quot;, root.right.left.data)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>root.right.left.data: E</code></pre></div></div>
<p class="v2-p"><strong>Why E?</strong></p>
<ul class="v2-ul"><li><code>root</code> is R</li><li><code>root.right</code> is B (R's right child)</li><li><code>root.right.left</code> is E (B's left child)</li><li><code>.data</code> gives us the value stored there: <code>E</code></li></ul>
<blockquote class="v2-bq"><p class="v2-p"><strong>Thinking Prompt:</strong> What would <code>root.right.right.left.data</code> print? Think about it before running the code!</p></blockquote>
<hr class="v2-hr">
<h3 class="v2-h3">Types of Binary Trees</h3>
<p class="v2-p">Before we move on, let us understand the different "shapes" a Binary Tree can have. These shapes matter because they affect performance.</p>
<h4 class="v2-h4">Balanced Binary Tree</h4>
<p class="v2-p">A <strong>balanced</strong> tree has at most 1 in height-difference between its left and right subtrees, for every node.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>        11
       /  \
      7    15
     / \  /  \
    3   9 13  19
               /
              18</code></pre></div></div>
<p class="v2-p">This tree is balanced · no side is much taller than the other.</p>
<h4 class="v2-h4">Complete Binary Tree</h4>
<p class="v2-p">A <strong>complete</strong> tree has all levels fully filled except possibly the last one, which is filled from left to right.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>        11
       /  \
      7    15
     / \  /  \
    3   9 13  19
   / \  /
  2   4  8</code></pre></div></div>
<p class="v2-p">All levels are full except the last, and the last fills left to right. A complete tree is also balanced.</p>
<h4 class="v2-h4">Full Binary Tree</h4>
<p class="v2-p">A <strong>full</strong> tree means every node has either 0 or 2 children · never just one.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>      11
     /  \
    7    15
        /  \
       13  19
      /  \
     12   14</code></pre></div></div>
<h4 class="v2-h4">Perfect Binary Tree</h4>
<p class="v2-p">A <strong>perfect</strong> tree has every level completely full of nodes, and all leaf nodes are at the same depth. A perfect tree is also full, complete, and balanced.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>        11
       /  \
      7    15
     / \  /  \
    3   9 13  19</code></pre></div></div>
<blockquote class="v2-bq"><p class="v2-p"><strong>Real-world tip:</strong> Balanced trees give you the best performance for searching and sorting. Unbalanced trees can become as slow as a Linked List. That is why AVL Trees (which we cover later) automatically stay balanced!</p></blockquote>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>: Binary Trees</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(4)">Phase Complete — Unlock Next ✓</button></div></div>
<div class="phase locked" id="phase5"><div class="ph"><div class="pn">Phase 5 of 10</div><div class="pt">Part 3: Binary Tree Traversal</div><div class="pc" id="chk5"></div></div><div class="pb2"><h3 class="v2-h3">What Is Traversal?</h3>
<p class="v2-p"><strong>Traversal</strong> means visiting every node in a tree exactly once, in some specific order. It is like taking a tour of every room in a building.</p>
<p class="v2-p">With a Linked List, traversal is simple · just go left to right. But with a tree, it branches! So we need rules about which direction to go first.</p>
<p class="v2-p">There are two main categories of traversal:</p>
<p class="v2-p"><strong>Breadth First Search (BFS):</strong> Visit all nodes on the same level (same depth) before going deeper. Like reading a book page by page.</p>
<p class="v2-p"><strong>Depth First Search (DFS):</strong> Go as deep as possible down one branch before exploring another branch. There are three types of DFS traversal:</p>
<ul class="v2-ul"><li><strong>Pre-order</strong></li><li><strong>In-order</strong></li><li><strong>Post-order</strong></li></ul>
<p class="v2-p">Let us explore each one using this example tree:</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>        R
       / \
      A   B
     / \ / \
    C  D E  F
              \
               G</code></pre></div></div>
<hr class="v2-hr">
<h3 class="v2-h3">Pre-order Traversal (Root → Left → Right)</h3>
<p class="v2-p"><strong>Pre-order</strong> visits the current node FIRST ("pre" = before), then goes left, then goes right.</p>
<p class="v2-p">Think of it like exploring a file system: you note down the folder name first, then explore inside it.</p>
<p class="v2-p"><strong>Order: Visit node → Go Left → Go Right</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>def preOrderTraversal(node):
    if node is None:       # If there is no node here, stop
        return
    print(node.data, end=&quot;, &quot;)       # 1. Visit (print) the current node
    preOrderTraversal(node.left)     # 2. Explore the left subtree
    preOrderTraversal(node.right)    # 3. Explore the right subtree

preOrderTraversal(root)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>R, A, C, D, B, E, F, G,</code></pre></div></div>
<p class="v2-p"><strong>How does it work?</strong></p>
<ol class="v2-ol"><li>Visit R → print "R"</li><li>Go left to A → print "A"</li><li>Go left to C → print "C"</li><li>C has no left child → return</li><li>C has no right child → return</li><li>Back at A → go right to D → print "D"</li><li>D has no children → return</li><li>Back at R → go right to B → print "B"</li><li>Go left to E → print "E"</li><li>E has no children → continue...</li><li>Go right to F → print "F"</li><li>F has left child G → print "G"</li></ol>
<blockquote class="v2-bq"><p class="v2-p"><strong>Real-world use:</strong> Pre-order is used to copy or serialize a tree (saving it to a file in the right order to rebuild it later).</p></blockquote>
<hr class="v2-hr">
<h3 class="v2-h3">In-order Traversal (Left → Root → Right)</h3>
<p class="v2-p"><strong>In-order</strong> visits the left subtree first, then the current node ("in" = the node is visited "in between" the left and right subtrees), then the right subtree.</p>
<p class="v2-p"><strong>Order: Go Left → Visit node → Go Right</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>def inOrderTraversal(node):
    if node is None:
        return
    inOrderTraversal(node.left)      # 1. Explore the left subtree first
    print(node.data, end=&quot;, &quot;)       # 2. Visit (print) the current node
    inOrderTraversal(node.right)     # 3. Explore the right subtree</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>C, A, D, R, E, B, G, F,</code></pre></div></div>
<blockquote class="v2-bq"><p class="v2-p"><strong>Very important fact:</strong> When you do in-order traversal on a <strong>Binary Search Tree</strong>, the values always come out in <strong>ascending (sorted) order</strong>! This is incredibly useful.</p></blockquote>
<hr class="v2-hr">
<h3 class="v2-h3">Post-order Traversal (Left → Right → Root)</h3>
<p class="v2-p"><strong>Post-order</strong> visits the left subtree, then the right subtree, and LAST ("post" = after) visits the current node.</p>
<p class="v2-p">Think of it like cleaning up a directory: you delete files inside folders before you delete the folder itself.</p>
<p class="v2-p"><strong>Order: Go Left → Go Right → Visit node</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>def postOrderTraversal(node):
    if node is None:
        return
    postOrderTraversal(node.left)    # 1. Explore the left subtree
    postOrderTraversal(node.right)   # 2. Explore the right subtree
    print(node.data, end=&quot;, &quot;)       # 3. Visit (print) the current node LAST</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>C, D, A, E, G, F, B, R,</code></pre></div></div>
<blockquote class="v2-bq"><p class="v2-p"><strong>Real-world use:</strong> Post-order is used when deleting a tree (delete children before the parent) or evaluating mathematical expression trees.</p></blockquote>
<hr class="v2-hr">
<h3 class="v2-h3">Quick Traversal Reference Table</h3>
<div class="v2-table-wrap"><table class="v2-table"><thead><tr><th>Traversal</th><th>Order</th><th>Output for our tree</th></tr></thead><tbody><tr><td>Pre-order</td><td>Root → Left → Right</td><td>R, A, C, D, B, E, F, G</td></tr><tr><td>In-order</td><td>Left → Root → Right</td><td>C, A, D, R, E, B, G, F</td></tr><tr><td>Post-order</td><td>Left → Right → Root</td><td>C, D, A, E, G, F, B, R</td></tr></tbody></table></div>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>: Binary Tree Traversal</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(5)">Phase Complete — Unlock Next ✓</button></div></div>
<div class="phase locked" id="phase6"><div class="ph"><div class="pn">Phase 6 of 10</div><div class="pt">Part 4: Binary Search Trees (BST)</div><div class="pc" id="chk6"></div></div><div class="pb2"><h3 class="v2-h3">What Is a Binary Search Tree?</h3>
<p class="v2-p">A <strong>Binary Search Tree (BST)</strong> is a special type of Binary Tree with one powerful rule:</p>
<blockquote class="v2-bq"><p class="v2-p">For every node X:</p>
<ul class="v2-ul"><li>All values in the <strong>left subtree</strong> are <strong>LESS than</strong> X</li><li>All values in the <strong>right subtree</strong> are <strong>GREATER than</strong> X</li><li>Both left and right subtrees must also follow this same rule</li></ul></blockquote>
<p class="v2-p">Think of it like a number-guessing game. Is the number you're looking for smaller or bigger than the current number? If smaller, go left. If bigger, go right. You eliminate half the possibilities each time!</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>           13
          /  \
         7    15
        / \  /  \
       3   8 14  19
                /
               18</code></pre></div></div>
<p class="v2-p"><strong>Check the rule:</strong></p>
<ul class="v2-ul"><li>Node 13: All left values (3, 7, 8) &lt; 13 ✓  All right values (14, 15, 18, 19) &gt; 13 ✓</li><li>Node 7: All left values (3) &lt; 7 ✓  All right values (8) &gt; 7 ✓</li><li>Node 15: Left (14) &lt; 15 ✓  Right (18, 19) &gt; 15 ✓</li></ul>
<h3 class="v2-h3">Key BST Terminology</h3>
<ul class="v2-ul"><li><strong>Size (n):</strong> Total number of nodes in the tree</li><li><strong>Height (h):</strong> Maximum number of edges from the root to a leaf</li><li><strong>In-order successor:</strong> The node that would come next if you did an in-order traversal. For node 13, the successor is 14 (the next higher value)</li></ul>
<hr class="v2-hr">
<h3 class="v2-h3">BST Traversal · Proving It Is a BST</h3>
<p class="v2-p">A quick way to verify that a Binary Tree is a BST: do an <strong>in-order traversal</strong> and check that the output is in ascending order.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>class TreeNode:
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None

def inOrderTraversal(node):
    if node is None:
        return
    inOrderTraversal(node.left)
    print(node.data, end=&quot;, &quot;)
    inOrderTraversal(node.right)

# Build the BST from the diagram above
root = TreeNode(13)
node7 = TreeNode(7)
node15 = TreeNode(15)
node3 = TreeNode(3)
node8 = TreeNode(8)
node14 = TreeNode(14)
node19 = TreeNode(19)
node18 = TreeNode(18)

root.left = node7
root.right = node15
node7.left = node3
node7.right = node8
node15.left = node14
node15.right = node19
node19.left = node18

# Traverse to verify
inOrderTraversal(root)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>3, 7, 8, 13, 14, 15, 18, 19,</code></pre></div></div>
<p class="v2-p">The output is in ascending order · confirming it is a valid BST!</p>
<hr class="v2-hr">
<h3 class="v2-h3">Searching in a BST</h3>
<p class="v2-p">Searching in a BST is fast because at every node, you can eliminate half the tree.</p>
<p class="v2-p"><strong>Algorithm:</strong></p>
<ol class="v2-ol"><li>Start at the root</li><li>If the current node equals the target → found it!</li><li>If the target is <strong>less than</strong> current node → search the <strong>left</strong> subtree</li><li>If the target is <strong>greater than</strong> current node → search the <strong>right</strong> subtree</li><li>If we reach <code>None</code> (an empty spot) → the value doesn't exist</li></ol>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>def search(node, target):
    if node is None:
        return None                        # Value not found
    elif node.data == target:
        return node                        # Found it!
    elif target &lt; node.data:
        return search(node.left, target)   # Go left (smaller values)
    else:
        return search(node.right, target)  # Go right (larger values)

# Search for the value 13
result = search(root, 13)
if result:
    print(f&quot;Found the node with value: {result.data}&quot;)
else:
    print(&quot;Value not found in the BST.&quot;)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Found the node with value: 13</code></pre></div></div>
<p class="v2-p"><strong>Time complexity:</strong> O(h), where h is the height of the tree. For a balanced tree, h ≈ log(n), which is very fast. For an unbalanced tree, h could be n (as slow as a list).</p>
<blockquote class="v2-bq"><p class="v2-p"><strong>Thinking Prompt:</strong> If you search for the value 10 in the BST above, which nodes does the algorithm visit? (Hint: it goes 13 → 7 → 8 → and then?)</p></blockquote>
<hr class="v2-hr">
<h3 class="v2-h3">Inserting Into a BST</h3>
<p class="v2-p">Inserting a new value works exactly like searching · but when you reach an empty spot, you place the new node there. Inserted nodes always become new leaf nodes.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>def insert(node, data):
    if node is None:
        return TreeNode(data)    # Create and return the new node here
    else:
        if data &lt; node.data:
            node.left = insert(node.left, data)    # Insert in left subtree
        elif data &gt; node.data:
            node.right = insert(node.right, data)  # Insert in right subtree
        # If data == node.data, we do nothing (no duplicates in BST)
    return node

# Insert the value 10
root = insert(root, 10)

# Verify by traversal — 10 should appear between 8 and 13
inOrderTraversal(root)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>3, 7, 8, 10, 13, 14, 15, 18, 19,</code></pre></div></div>
<p class="v2-p">The value 10 was correctly placed between 8 and 13.</p>
<hr class="v2-hr">
<h3 class="v2-h3">Finding the Minimum Value in a Subtree</h3>
<p class="v2-p">Before we cover deletion, we need a helper: finding the smallest value in a subtree.</p>
<p class="v2-p"><strong>The rule is simple:</strong> In a BST, the smallest value is always the <strong>leftmost node</strong>. Just keep going left until you can't anymore.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>def minValueNode(node):
    current = node
    while current.left is not None:    # Keep going left
        current = current.left
    return current                     # This is the minimum

print(&quot;Lowest value:&quot;, minValueNode(root).data)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Lowest value: 3</code></pre></div></div>
<p class="v2-p">We use this function during deletion to find a node's <strong>in-order successor</strong> (the smallest value in the right subtree).</p>
<hr class="v2-hr">
<h3 class="v2-h3">Deleting a Node in a BST</h3>
<p class="v2-p">Deletion is the most complex BST operation. There are three cases:</p>
<p class="v2-p"><strong>Case 1: The node to delete is a leaf</strong> (no children) → Simply remove it. Set the parent's pointer to <code>None</code>.</p>
<p class="v2-p"><strong>Case 2: The node has ONE child</strong> → Replace the node with its one child. The parent now points to the grandchild.</p>
<p class="v2-p"><strong>Case 3: The node has TWO children</strong> → Find the node's <strong>in-order successor</strong> (smallest value in the right subtree). Copy that value into the current node. Then delete the successor (which will be Case 1 or Case 2).</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>def delete(node, data):
    if node is None:
        return None

    if data &lt; node.data:
        node.left = delete(node.left, data)       # Go left
    elif data &gt; node.data:
        node.right = delete(node.right, data)     # Go right
    else:
        # Found the node to delete!

        # Case 1 &amp; 2: Zero or one child
        if node.left is None:
            return node.right        # Replace with right child (or None)
        elif node.right is None:
            return node.left         # Replace with left child

        # Case 3: Two children
        # Find in-order successor (smallest in right subtree)
        successor = minValueNode(node.right)
        # Copy successor&#x27;s value to this node
        node.data = successor.data
        # Delete the successor from the right subtree
        node.right = delete(node.right, successor.data)

    return node

# Delete node 15 (which has two children: 14 and 19)
root = delete(root, 15)
inOrderTraversal(root)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>3, 7, 8, 10, 13, 14, 18, 19,</code></pre></div></div>
<p class="v2-p">Node 15 was replaced by 18 (its in-order successor · the smallest value in its right subtree), and then the old 18 was deleted.</p>
<hr class="v2-hr">
<h3 class="v2-h3">Balanced vs. Unbalanced BSTs</h3>
<p class="v2-p">Here is a critical issue with BSTs: if you insert values in sorted order, the tree becomes "lopsided" (unbalanced), and performance degrades to O(n) · as slow as a linked list.</p>
<p class="v2-p"><strong>Balanced BST (good):</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>         13
        /   \
       7     15
      / \   /  \
     3   8 14   19
              /
             18</code></pre></div></div>
<p class="v2-p">Height = 4. Searching takes at most 4 steps.</p>
<p class="v2-p"><strong>Unbalanced BST (bad · if you insert 7, 13, 15, 19 in order):</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>7
 \
  13
    \
     15
       \
        19</code></pre></div></div>
<p class="v2-p">Height = 4. But with 1000 nodes, you might need 1000 steps to find anything!</p>
<p class="v2-p">This is exactly the problem that <strong>AVL Trees</strong> solve, coming up next.</p>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>: Binary Search Trees (BST)</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(6)">Phase Complete — Unlock Next ✓</button></div></div>
<div class="phase locked" id="phase7"><div class="ph"><div class="pn">Phase 7 of 10</div><div class="pt">Part 5: AVL Trees · Self-Balancing Trees</div><div class="pc" id="chk7"></div></div><div class="pb2"><h3 class="v2-h3">What Is an AVL Tree?</h3>
<p class="v2-p">An <strong>AVL Tree</strong> is a Binary Search Tree that <strong>automatically keeps itself balanced</strong> after every insertion or deletion.</p>
<p class="v2-p">The name "AVL" comes from two Soviet inventors: Georgy <strong>A</strong>delson-<strong>V</strong>elsky and Evgenii <strong>L</strong>andis, who invented it in 1962.</p>
<p class="v2-p">Why does this matter? Because an AVL Tree <strong>guarantees</strong> that all operations (search, insert, delete) run in <strong>O(log n)</strong> time · even in the worst case. Regular BSTs can degrade to O(n) if the data is inserted in sorted order.</p>
<p class="v2-p">The only difference between a regular BST and an AVL Tree is that the AVL Tree performs <strong>rotation operations</strong> to restore balance whenever it becomes unbalanced.</p>
<p class="v2-p"><strong>Unbalanced BST (height 6):</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>B
 \
  G
   \
    E
     \
      K
       \
        F
         \
          P
           \
            I
             \
              M</code></pre></div></div>
<p class="v2-p"><strong>Same data as an AVL Tree (height 3):</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>       G
      / \
     E   K
    / \ / \
   B  F I  P
           /
          M</code></pre></div></div>
<p class="v2-p">Both trees contain the same data, but the AVL Tree's height is dramatically smaller!</p>
<hr class="v2-hr">
<h3 class="v2-h3">The Balance Factor</h3>
<p class="v2-p">Every node in an AVL Tree keeps track of its <strong>Balance Factor (BF)</strong>: the difference between the heights of its right and left subtrees.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>BF(X) = height(right subtree of X) - height(left subtree of X)</code></pre></div></div>
<p class="v2-p"><strong>Interpreting the balance factor:</strong></p>
<ul class="v2-ul"><li><strong>BF = 0:</strong> The node is perfectly balanced</li><li><strong>BF = +1:</strong> The right side is one taller (acceptable)</li><li><strong>BF = -1:</strong> The left side is one taller (acceptable)</li><li><strong>BF ≥ +2:</strong> Too right-heavy → must rotate to fix!</li><li><strong>BF ≤ -2:</strong> Too left-heavy → must rotate to fix!</li></ul>
<p class="v2-p">An AVL Tree checks and updates balance factors after every insert or delete. If any node has a balance factor outside [-1, 1], rotations are performed.</p>
<hr class="v2-hr">
<h3 class="v2-h3">Left and Right Rotations</h3>
<p class="v2-p"><strong>Rotations</strong> are the mechanism AVL Trees use to fix imbalance. They rearrange nodes while keeping the BST property (left &lt; parent &lt; right) intact.</p>
<p class="v2-p">Think of a rotation like a "pivot": you take a node and rotate the tree around it.</p>
<h4 class="v2-h4">Right Rotation (fixes left-heavy imbalance)</h4>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Before (left-heavy, BF = -2):
      Z
     /
    Y
   /
  X

After right rotation at Z:
    Y
   / \
  X   Z</code></pre></div></div>
<h4 class="v2-h4">Left Rotation (fixes right-heavy imbalance)</h4>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Before (right-heavy, BF = +2):
  X
   \
    Y
     \
      Z

After left rotation at X:
    Y
   / \
  X   Z</code></pre></div></div>
<hr class="v2-hr">
<h3 class="v2-h3">The Four Out-of-Balance Cases</h3>
<p class="v2-p">When a tree becomes unbalanced, there are exactly <strong>four cases</strong>, each requiring a specific fix:</p>
<div class="v2-table-wrap"><table class="v2-table"><thead><tr><th>Case</th><th>Description</th><th>Fix</th></tr></thead><tbody><tr><td><strong>Left-Left (LL)</strong></td><td>Unbalanced node AND its left child are both left-heavy (BF negative)</td><td>Single <strong>right</strong> rotation</td></tr><tr><td><strong>Right-Right (RR)</strong></td><td>Unbalanced node AND its right child are both right-heavy (BF positive)</td><td>Single <strong>left</strong> rotation</td></tr><tr><td><strong>Left-Right (LR)</strong></td><td>Unbalanced node is left-heavy, but its left child is right-heavy</td><td>First <strong>left rotate</strong> the left child, then <strong>right rotate</strong> the unbalanced node</td></tr><tr><td><strong>Right-Left (RL)</strong></td><td>Unbalanced node is right-heavy, but its right child is left-heavy</td><td>First <strong>right rotate</strong> the right child, then <strong>left rotate</strong> the unbalanced node</td></tr></tbody></table></div>
<p class="v2-p"><strong>Simple way to remember:</strong> If both the problem node and its heavy child lean the same direction → single rotation. If they lean in opposite directions → double rotation (fix the child first, then the node).</p>
<hr class="v2-hr">
<h3 class="v2-h3">AVL Tree Implementation in Python</h3>
<p class="v2-p">Now let's put it all together. This is a complete AVL Tree implementation with insertion, deletion, and automatic rebalancing.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>class AVLNode:
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None
        self.height = 1    # NEW: AVL nodes track their own height

class AVLTree:

    def getHeight(self, node):
        &quot;&quot;&quot;Returns the height of a node (0 if None).&quot;&quot;&quot;
        if node is None:
            return 0
        return node.height

    def getBalanceFactor(self, node):
        &quot;&quot;&quot;Calculates the balance factor of a node.&quot;&quot;&quot;
        if node is None:
            return 0
        return self.getHeight(node.right) - self.getHeight(node.left)

    def updateHeight(self, node):
        &quot;&quot;&quot;Recalculates height from children.&quot;&quot;&quot;
        node.height = 1 + max(self.getHeight(node.left), self.getHeight(node.right))

    def rotateRight(self, z):
        &quot;&quot;&quot;Performs a right rotation around node z.&quot;&quot;&quot;
        y = z.left          # y is the left child of z
        T3 = y.right        # T3 is y&#x27;s right subtree

        # Perform the rotation
        y.right = z         # z becomes y&#x27;s right child
        z.left = T3         # T3 moves to be z&#x27;s left child

        # Update heights (z first, then y, since y is now higher)
        self.updateHeight(z)
        self.updateHeight(y)

        return y            # y is the new root of this subtree

    def rotateLeft(self, z):
        &quot;&quot;&quot;Performs a left rotation around node z.&quot;&quot;&quot;
        y = z.right         # y is the right child of z
        T2 = y.left         # T2 is y&#x27;s left subtree

        # Perform the rotation
        y.left = z          # z becomes y&#x27;s left child
        z.right = T2        # T2 moves to be z&#x27;s right child

        # Update heights
        self.updateHeight(z)
        self.updateHeight(y)

        return y            # y is the new root of this subtree

    def rebalance(self, node):
        &quot;&quot;&quot;Checks balance and rotates if needed.&quot;&quot;&quot;
        self.updateHeight(node)
        bf = self.getBalanceFactor(node)

        # Left-Left case (too left-heavy, left child is also left-heavy)
        if bf &lt; -1 and self.getBalanceFactor(node.left) &lt;= 0:
            return self.rotateRight(node)

        # Right-Right case (too right-heavy, right child is also right-heavy)
        if bf &gt; 1 and self.getBalanceFactor(node.right) &gt;= 0:
            return self.rotateLeft(node)

        # Left-Right case (left-heavy, but left child is right-heavy)
        if bf &lt; -1 and self.getBalanceFactor(node.left) &gt; 0:
            node.left = self.rotateLeft(node.left)   # Fix child first
            return self.rotateRight(node)

        # Right-Left case (right-heavy, but right child is left-heavy)
        if bf &gt; 1 and self.getBalanceFactor(node.right) &lt; 0:
            node.right = self.rotateRight(node.right) # Fix child first
            return self.rotateLeft(node)

        return node  # Already balanced

    def insert(self, node, data):
        &quot;&quot;&quot;Inserts data like a regular BST, then rebalances.&quot;&quot;&quot;
        if node is None:
            return AVLNode(data)
        if data &lt; node.data:
            node.left = self.insert(node.left, data)
        elif data &gt; node.data:
            node.right = self.insert(node.right, data)
        else:
            return node   # Duplicate values not allowed

        return self.rebalance(node)

    def inOrder(self, node):
        &quot;&quot;&quot;In-order traversal to verify BST order.&quot;&quot;&quot;
        if node is None:
            return
        self.inOrder(node.left)
        print(node.data, end=&quot;, &quot;)
        self.inOrder(node.right)


# Demo: Build an AVL tree that would be unbalanced as a regular BST
avl = AVLTree()
root = None

# Insert values in sorted order — a regular BST would become a straight line
for val in [10, 20, 30, 40, 50, 25]:
    root = avl.insert(root, val)

print(&quot;In-order traversal (should be sorted):&quot;)
avl.inOrder(root)
print()
print(&quot;Root value (should be near middle due to balancing):&quot;, root.data)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>In-order traversal (should be sorted):
10, 20, 25, 30, 40, 50,
Root value (should be near middle due to balancing): 30</code></pre></div></div>
<p class="v2-p">Even though we inserted values 10 → 20 → 30 → 40 → 50 → 25 in an order that would make a regular BST into a long chain, the AVL Tree kept itself balanced, with 30 at the root!</p>
<hr class="v2-hr">
<h3 class="v2-h3">Retracing After Insertion or Deletion</h3>
<p class="v2-p">After every insert or delete, the AVL Tree "retraces" its steps back up to the root, updating heights and fixing any imbalances it finds along the way. This is handled automatically through recursion.</p>
<p class="v2-p">If node H is unbalanced, the algorithm fixes H first. After the rotation at H, nodes higher up (like E and C) often find they are now balanced too · so fixing one often fixes many.</p>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>: AVL Trees · Self-Balancing Trees</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(7)">Phase Complete — Unlock Next ✓</button></div></div>
<div class="phase locked" id="phase8"><div class="ph"><div class="pn">Phase 8 of 10</div><div class="pt">Part 6: Graphs</div><div class="pc" id="chk8"></div></div><div class="pb2"><h3 class="v2-h3">What Is a Graph?</h3>
<p class="v2-p">A <strong>Graph</strong> is a data structure made of <strong>vertices</strong> (also called nodes) and <strong>edges</strong> (connections between vertices).</p>
<p class="v2-p">Think of a graph like a city map: the cities are vertices, and the roads connecting them are edges. Or think of social media: each person is a vertex, and friendships are edges.</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>    F
   2|4
  B   C
  |     \
  A   E  D
  |    \ /
  G     D</code></pre></div></div>
<p class="v2-p">Unlike a Tree, a Graph has <strong>no root</strong> and <strong>no parent-child hierarchy</strong>. Any vertex can connect to any other vertex. There can even be cycles (paths that loop back to where you started).</p>
<h3 class="v2-h3">Why Are Graphs So Important?</h3>
<p class="v2-p">Graphs model almost every real-world network:</p>
<ul class="v2-ul"><li><strong>Social networks:</strong> Facebook, Instagram, LinkedIn · people (vertices) connected by friendships (edges)</li><li><strong>Maps and GPS:</strong> Locations (vertices) connected by roads (edges). Shortest path algorithms find the fastest route</li><li><strong>The internet:</strong> Web pages (vertices) connected by hyperlinks (edges)</li><li><strong>Biology:</strong> Neural networks, protein interactions, disease spread</li><li><strong>Logistics:</strong> Delivery networks, supply chains</li></ul>
<h3 class="v2-h3">Key Graph Vocabulary</h3>
<div class="v2-table-wrap"><table class="v2-table"><thead><tr><th>Term</th><th>Meaning</th></tr></thead><tbody><tr><td><strong>Vertex (Node)</strong></td><td>A point/object in the graph</td></tr><tr><td><strong>Edge</strong></td><td>A connection between two vertices</td></tr><tr><td><strong>Adjacent (Neighbors)</strong></td><td>Two vertices that share an edge</td></tr><tr><td><strong>Directed Graph</strong></td><td>Edges have a direction (one-way roads)</td></tr><tr><td><strong>Undirected Graph</strong></td><td>Edges go both ways (two-way roads)</td></tr><tr><td><strong>Weighted Graph</strong></td><td>Edges have weights/costs (like road distances)</td></tr></tbody></table></div>
<hr class="v2-hr">
<h3 class="v2-h3">Graph Representations · How to Store a Graph in Python</h3>
<p class="v2-p">A Graph is not as simple to store in memory as a list or tree. We need a special structure. There are two main ways:</p>
<hr class="v2-hr">
<h4 class="v2-h4">Method 1: Adjacency Matrix</h4>
<p class="v2-p">An <strong>Adjacency Matrix</strong> is a 2D table (grid) where the row number represents the "from" vertex and the column number represents the "to" vertex.</p>
<p class="v2-p">If there is an edge from vertex <code>i</code> to vertex <code>j</code>, we put a <code>1</code> (or the edge weight) in cell <code>[i][j]</code>.</p>
<p class="v2-p"><strong>Example: Simple undirected graph with 4 vertices A, B, C, D</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Edges: A-B, A-C, A-D, B-C</code></pre></div></div>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>   A  B  C  D
A [0, 1, 1, 1]
B [1, 0, 1, 0]
C [1, 1, 0, 0]
D [1, 0, 0, 0]</code></pre></div></div>
<p class="v2-p">Notice the matrix is <strong>symmetric</strong> for undirected graphs (if A connects to B, then B connects to A).</p>
<p class="v2-p"><strong>For a weighted, directed graph</strong> (e.g., A→B with weight 3, A→D with weight 2, B→C with weight 1, D→A with weight 4):</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>   A  B  C  D
A [0, 3, 0, 2]
B [0, 0, 1, 0]
C [0, 0, 0, 0]
D [4, 0, 0, 0]</code></pre></div></div>
<p class="v2-p"><strong>Python implementation:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code># Adjacency matrix for an undirected graph
# Vertices: A=0, B=1, C=2, D=3

size = 4
adj_matrix = [[0] * size for _ in range(size)]   # Create a 4x4 grid of zeros

def add_edge(matrix, u, v, weight=1, directed=False):
    &quot;&quot;&quot;Add an edge from vertex u to vertex v.&quot;&quot;&quot;
    matrix[u][v] = weight
    if not directed:
        matrix[v][u] = weight    # For undirected, add both directions

# Add edges (using index numbers: A=0, B=1, C=2, D=3)
add_edge(adj_matrix, 0, 1)   # A-B
add_edge(adj_matrix, 0, 2)   # A-C
add_edge(adj_matrix, 0, 3)   # A-D
add_edge(adj_matrix, 1, 2)   # B-C

# Print the adjacency matrix
print(&quot;Adjacency Matrix:&quot;)
print(&quot;   A  B  C  D&quot;)
labels = [&#x27;A&#x27;, &#x27;B&#x27;, &#x27;C&#x27;, &#x27;D&#x27;]
for i, row in enumerate(adj_matrix):
    print(f&quot;{labels[i]} {row}&quot;)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Adjacency Matrix:
   A  B  C  D
A [0, 1, 1, 1]
B [1, 0, 1, 0]
C [1, 1, 0, 0]
D [1, 0, 0, 0]</code></pre></div></div>
<p class="v2-p"><strong>Pros of Adjacency Matrix:</strong> Simple to implement. Checking if an edge exists between two specific vertices is O(1) · instant! <strong>Cons:</strong> Uses a LOT of memory if the graph has many vertices but few edges (most cells will be 0).</p>
<hr class="v2-hr">
<h4 class="v2-h4">Method 2: Adjacency List</h4>
<p class="v2-p">An <strong>Adjacency List</strong> stores each vertex with a list of all its neighbors. This is much more memory-efficient for "sparse" graphs (graphs where most vertices are not connected to each other).</p>
<p class="v2-p"><strong>Python implementation:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code># Adjacency list for the same graph using a dictionary
adj_list = {
    &#x27;A&#x27;: [&#x27;B&#x27;, &#x27;C&#x27;, &#x27;D&#x27;],
    &#x27;B&#x27;: [&#x27;A&#x27;, &#x27;C&#x27;],
    &#x27;C&#x27;: [&#x27;A&#x27;, &#x27;B&#x27;],
    &#x27;D&#x27;: [&#x27;A&#x27;]
}

def add_edge_list(graph, u, v, directed=False):
    &quot;&quot;&quot;Add an edge to the adjacency list.&quot;&quot;&quot;
    if u not in graph:
        graph[u] = []
    graph[u].append(v)
    if not directed:
        if v not in graph:
            graph[v] = []
        graph[v].append(u)

# Build a new graph
graph = {}
add_edge_list(graph, &#x27;A&#x27;, &#x27;B&#x27;)
add_edge_list(graph, &#x27;A&#x27;, &#x27;C&#x27;)
add_edge_list(graph, &#x27;A&#x27;, &#x27;D&#x27;)
add_edge_list(graph, &#x27;B&#x27;, &#x27;C&#x27;)

print(&quot;Adjacency List:&quot;)
for vertex, neighbors in graph.items():
    print(f&quot;  {vertex}: {neighbors}&quot;)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Adjacency List:
  A: [&#x27;B&#x27;, &#x27;C&#x27;, &#x27;D&#x27;]
  B: [&#x27;A&#x27;, &#x27;C&#x27;]
  C: [&#x27;A&#x27;, &#x27;B&#x27;]
  D: [&#x27;A&#x27;]</code></pre></div></div>
<p class="v2-p"><strong>For a directed, weighted graph</strong> · store tuples <code>(neighbor, weight)</code>:</p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>weighted_directed = {
    &#x27;A&#x27;: [(&#x27;B&#x27;, 3), (&#x27;D&#x27;, 2)],   # A→B with weight 3, A→D with weight 2
    &#x27;B&#x27;: [(&#x27;C&#x27;, 1)],              # B→C with weight 1
    &#x27;C&#x27;: [],                       # C has no outgoing edges
    &#x27;D&#x27;: [(&#x27;A&#x27;, 4)]               # D→A with weight 4
}

print(&quot;Weighted Directed Adjacency List:&quot;)
for vertex, edges in weighted_directed.items():
    print(f&quot;  {vertex}: {edges}&quot;)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>Weighted Directed Adjacency List:
  A: [(&#x27;B&#x27;, 3), (&#x27;D&#x27;, 2)]
  B: [(&#x27;C&#x27;, 1)]
  C: []
  D: [(&#x27;A&#x27;, 4)]</code></pre></div></div>
<p class="v2-p"><strong>Pros of Adjacency List:</strong> Memory-efficient for sparse graphs. Easy to iterate over all neighbors of a vertex. <strong>Cons:</strong> Checking if a specific edge exists requires scanning the neighbor list · O(degree) time.</p>
<hr class="v2-hr">
<h3 class="v2-h3">Which Representation Should You Use?</h3>
<div class="v2-table-wrap"><table class="v2-table"><thead><tr><th>Situation</th><th>Recommended Representation</th></tr></thead><tbody><tr><td>Dense graph (many edges)</td><td>Adjacency Matrix</td></tr><tr><td>Sparse graph (few edges)</td><td>Adjacency List</td></tr><tr><td>Need to quickly check if edge (u,v) exists</td><td>Adjacency Matrix</td></tr><tr><td>Need to iterate over all neighbors of a vertex</td><td>Adjacency List</td></tr><tr><td>This tutorial going forward</td><td><strong>Adjacency Matrix</strong> (simpler to understand)</td></tr></tbody></table></div>
<hr class="v2-hr"><div class="task-box"><div class="task-lbl">✏️ Your Task</div><div class="task-body">Practise what you just learned about <strong>: Graphs</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="ub" onclick="unlockNext(8)">Phase Complete — Unlock Next ✓</button></div></div>
<div class="phase locked" id="phase9"><div class="ph"><div class="pn">Phase 9 of 10</div><div class="pt">Part 7: Guided Practice Exercises</div><div class="pc" id="chk9"></div></div><div class="pb2"><div class="chal-box"><div class="chal-lbl">🎯 Your Challenge</div><div class="chal-body"><h3 class="v2-h3">Exercise 1: Build and Traverse a BST</h3>
<p class="v2-p"><strong>Objective:</strong> Create a Binary Search Tree and verify it using in-order traversal.</p>
<p class="v2-p"><strong>Scenario:</strong> You are building a student grade lookup system. Each student has an ID number. You want to store them in a BST for fast lookup.</p>
<p class="v2-p"><strong>Steps:</strong></p>
<ol class="v2-ol"><li>Create a <code>TreeNode</code> class with <code>data</code>, <code>left</code>, and <code>right</code></li><li>Create an <code>insert()</code> function</li><li>Insert these student IDs in this order: <code>50, 30, 70, 20, 40, 60, 80</code></li><li>Perform in-order traversal to verify the tree is a valid BST</li></ol></div></div><div class="task-box"><div class="task-lbl">✏️ Task</div><div class="task-body">Practise what you just learned about <strong>: Guided Practice Exercises</strong>. Open your editor, type the examples above by hand, modify them, and observe what changes.</div></div><button class="reveal-btn" onclick="toggleReveal(this)">Reveal Answer 👁️</button><div class="reveal-content"><div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">python</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>class TreeNode:
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None

def insert(node, data):
    if node is None:
        return TreeNode(data)
    if data &lt; node.data:
        node.left = insert(node.left, data)
    elif data &gt; node.data:
        node.right = insert(node.right, data)
    return node

def inOrder(node):
    if node is None:
        return
    inOrder(node.left)
    print(node.data, end=&quot;, &quot;)
    inOrder(node.right)

# Build the BST
root = None
for val in [50, 30, 70, 20, 40, 60, 80]:
    root = insert(root, val)

print(&quot;In-order traversal:&quot;)
inOrder(root)</code></pre></div></div>
<p class="v2-p"><strong>Expected Output:</strong></p>
<div class="v2-code-wrap"><div class="v2-code-bar"><span class="v2-dot r"></span><span class="v2-dot y"></span><span class="v2-dot g"></span><span class="v2-code-lang">code</span><button class="copy-btn" onclick="copyCode(this)">&#10064; Copy</button></div><div class="v2-code-body"><pre><code>In-order traversal:
20, 30, 40, 50, 60, 70, 80,</code></pre></div></div>