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<!-- ====== PAGE HEADER ====== -->
<div class="page-header">
<div class="breadcrumb"><a href="index.html">Home</a> / Trees & Binary Search Trees</div>
<h1>Trees & Binary Search Trees</h1>
<p>Hierarchical data structures that power databases, file systems, and countless interview problems.</p>
</div>
<!-- ====== TABLE OF CONTENTS ====== -->
<div class="toc">
<h4>Table of Contents</h4>
<a href="#what-is-a-tree">1. What is a Tree?</a>
<a href="#bst">2. Binary Search Tree (BST)</a>
<a href="#bst-implementation">3. BST Implementation</a>
<a href="#traversals">4. Tree Traversals</a>
<a href="#common-problems">5. Common Tree Problems & Patterns</a>
<a href="#leetcode">6. LeetCode Problems</a>
<a href="#balanced-trees">7. Balanced Trees</a>
<a href="#quiz">8. Practice Quiz</a>
</div>
<!-- ============================================================ -->
<!-- SECTION 1: WHAT IS A TREE? -->
<!-- ============================================================ -->
<section id="what-is-a-tree">
<h2>1. What is a Tree?</h2>
<p>A <strong>tree</strong> is a hierarchical data structure consisting of nodes connected by edges. Unlike arrays or linked lists which are linear, trees branch out -- making them perfect for representing hierarchical relationships like file systems, HTML DOM, or organizational charts.</p>
<h3>Key Terminology</h3>
<ul>
<li><strong>Root</strong> -- The topmost node (no parent)</li>
<li><strong>Child</strong> -- A node directly connected below another node</li>
<li><strong>Parent</strong> -- A node directly connected above another node</li>
<li><strong>Leaf</strong> -- A node with no children</li>
<li><strong>Depth</strong> -- Distance from the root to a node (root has depth 0)</li>
<li><strong>Height</strong> -- Distance from a node to its deepest leaf (leaves have height 0)</li>
<li><strong>Subtree</strong> -- A node and all its descendants</li>
<li><strong>Edge</strong> -- Connection between a parent and child</li>
</ul>
<h3>Visual Representation</h3>
<pre><code>
<span class="number">1</span> <-- Root (depth 0, height 2)
/ \
<span class="number">2</span> <span class="number">3</span> <-- depth 1
/ \ \
<span class="number">4</span> <span class="number">5</span> <span class="number">6</span> <-- Leaves (depth 2, height 0)
Nodes: 6
Edges: 5 (always n-1 for n nodes)
Height of tree: 2
</code></pre>
<h3>Binary Tree</h3>
<p>A <strong>binary tree</strong> is a tree where each node has <strong>at most 2 children</strong> -- referred to as the <strong>left child</strong> and <strong>right child</strong>. This is the most common tree type in interviews.</p>
<div class="formula-box">
Properties of Binary Trees:<br>
- A tree with n nodes has exactly n - 1 edges<br>
- Maximum nodes at depth d = 2^d<br>
- Maximum total nodes with height h = 2^(h+1) - 1<br>
- A complete binary tree of n nodes has height = floor(log2(n))
</div>
<div class="tip-box">
<div class="label">Tip</div>
<p>Trees are <strong>recursive</strong> by nature: every subtree is itself a tree. This is why most tree problems are elegantly solved with recursion.</p>
</div>
</section>
<!-- ============================================================ -->
<!-- SECTION 2: BINARY SEARCH TREE -->
<!-- ============================================================ -->
<section id="bst">
<h2>2. Binary Search Tree (BST)</h2>
<p>A <strong>Binary Search Tree</strong> is a binary tree with an ordering property: for every node, <strong>all values in its left subtree are less than the node's value</strong>, and <strong>all values in its right subtree are greater</strong>.</p>
<div class="formula-box">
BST Property: left < parent < right (for every node)
</div>
<h3>BST Example</h3>
<pre><code>
<span class="number">8</span>
/ \
<span class="number">3</span> <span class="number">10</span>
/ \ \
<span class="number">1</span> <span class="number">6</span> <span class="number">14</span>
/ \ /
<span class="number">4</span> <span class="number">7</span> <span class="number">13</span>
Inorder traversal gives sorted order:
1, 3, 4, 6, 7, 8, 10, 13, 14
</code></pre>
<h3>BST Operations & Time Complexity</h3>
<table>
<thead>
<tr>
<th>Operation</th>
<th>Average Case</th>
<th>Worst Case</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td>Search</td>
<td>O(log n)</td>
<td>O(n)</td>
<td>Find a value in the tree</td>
</tr>
<tr>
<td>Insert</td>
<td>O(log n)</td>
<td>O(n)</td>
<td>Add a new value</td>
</tr>
<tr>
<td>Delete</td>
<td>O(log n)</td>
<td>O(n)</td>
<td>Remove a value</td>
</tr>
<tr>
<td>Find Min/Max</td>
<td>O(log n)</td>
<td>O(n)</td>
<td>Go left/right until leaf</td>
</tr>
</tbody>
</table>
<p>All operations are <strong>O(h)</strong> where h is the height. For a <strong>balanced</strong> tree, h = O(log n). For a <strong>skewed</strong> tree (essentially a linked list), h = O(n).</p>
<div class="warning-box">
<div class="label">Worst Case</div>
<p>If you insert sorted data [1, 2, 3, 4, 5] into a BST, you get a skewed tree that looks like a linked list. Every operation becomes O(n). This is why balanced BSTs (AVL, Red-Black) exist.</p>
<pre><code> <span class="number">1</span>
\
<span class="number">2</span>
\
<span class="number">3</span> <-- Skewed tree: height = n - 1
\
<span class="number">4</span>
\
<span class="number">5</span></code></pre>
</div>
</section>
<!-- ============================================================ -->
<!-- SECTION 3: BST IMPLEMENTATION -->
<!-- ============================================================ -->
<section id="bst-implementation">
<h2>3. BST Implementation</h2>
<h3>TreeNode Class</h3>
<pre><code><span class="lang-label">Python</span>
<span class="keyword">class</span> <span class="function">TreeNode</span>:
<span class="keyword">def</span> <span class="function">__init__</span>(<span class="builtin">self</span>, val=<span class="number">0</span>):
<span class="builtin">self</span>.val = val
<span class="builtin">self</span>.left = <span class="keyword">None</span>
<span class="builtin">self</span>.right = <span class="keyword">None</span>
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="keyword">class</span> <span class="function">TreeNode</span> {
<span class="function">constructor</span>(val = <span class="number">0</span>) {
<span class="keyword">this</span>.val = val;
<span class="keyword">this</span>.left = <span class="keyword">null</span>;
<span class="keyword">this</span>.right = <span class="keyword">null</span>;
}
}
</code></pre>
<h3>Full BST Class -- Python</h3>
<pre><code><span class="lang-label">Python</span>
<span class="keyword">class</span> <span class="function">BST</span>:
<span class="keyword">def</span> <span class="function">__init__</span>(<span class="builtin">self</span>):
<span class="builtin">self</span>.root = <span class="keyword">None</span>
<span class="comment"># ---- INSERT ----</span>
<span class="keyword">def</span> <span class="function">insert</span>(<span class="builtin">self</span>, val):
<span class="builtin">self</span>.root = <span class="builtin">self</span>._insert(<span class="builtin">self</span>.root, val)
<span class="keyword">def</span> <span class="function">_insert</span>(<span class="builtin">self</span>, node, val):
<span class="keyword">if not</span> node:
<span class="keyword">return</span> <span class="function">TreeNode</span>(val)
<span class="keyword">if</span> val < node.val:
node.left = <span class="builtin">self</span>._insert(node.left, val)
<span class="keyword">elif</span> val > node.val:
node.right = <span class="builtin">self</span>._insert(node.right, val)
<span class="comment"># if val == node.val, ignore duplicate</span>
<span class="keyword">return</span> node
<span class="comment"># ---- SEARCH ----</span>
<span class="keyword">def</span> <span class="function">search</span>(<span class="builtin">self</span>, val):
<span class="keyword">return</span> <span class="builtin">self</span>._search(<span class="builtin">self</span>.root, val)
<span class="keyword">def</span> <span class="function">_search</span>(<span class="builtin">self</span>, node, val):
<span class="keyword">if not</span> node:
<span class="keyword">return</span> <span class="keyword">False</span>
<span class="keyword">if</span> val == node.val:
<span class="keyword">return</span> <span class="keyword">True</span>
<span class="keyword">elif</span> val < node.val:
<span class="keyword">return</span> <span class="builtin">self</span>._search(node.left, val)
<span class="keyword">else</span>:
<span class="keyword">return</span> <span class="builtin">self</span>._search(node.right, val)
<span class="comment"># ---- FIND MIN / MAX ----</span>
<span class="keyword">def</span> <span class="function">find_min</span>(<span class="builtin">self</span>, node=<span class="keyword">None</span>):
node = node <span class="keyword">or</span> <span class="builtin">self</span>.root
<span class="keyword">while</span> node.left:
node = node.left
<span class="keyword">return</span> node.val
<span class="keyword">def</span> <span class="function">find_max</span>(<span class="builtin">self</span>, node=<span class="keyword">None</span>):
node = node <span class="keyword">or</span> <span class="builtin">self</span>.root
<span class="keyword">while</span> node.right:
node = node.right
<span class="keyword">return</span> node.val
<span class="comment"># ---- DELETE ----</span>
<span class="keyword">def</span> <span class="function">delete</span>(<span class="builtin">self</span>, val):
<span class="builtin">self</span>.root = <span class="builtin">self</span>._delete(<span class="builtin">self</span>.root, val)
<span class="keyword">def</span> <span class="function">_delete</span>(<span class="builtin">self</span>, node, val):
<span class="keyword">if not</span> node:
<span class="keyword">return</span> <span class="keyword">None</span>
<span class="keyword">if</span> val < node.val:
node.left = <span class="builtin">self</span>._delete(node.left, val)
<span class="keyword">elif</span> val > node.val:
node.right = <span class="builtin">self</span>._delete(node.right, val)
<span class="keyword">else</span>:
<span class="comment"># Case 1: Leaf node (no children)</span>
<span class="keyword">if not</span> node.left <span class="keyword">and not</span> node.right:
<span class="keyword">return</span> <span class="keyword">None</span>
<span class="comment"># Case 2: One child</span>
<span class="keyword">if not</span> node.left:
<span class="keyword">return</span> node.right
<span class="keyword">if not</span> node.right:
<span class="keyword">return</span> node.left
<span class="comment"># Case 3: Two children</span>
<span class="comment"># Find inorder successor (smallest in right subtree)</span>
successor_val = <span class="builtin">self</span>.find_min(node.right)
node.val = successor_val
node.right = <span class="builtin">self</span>._delete(node.right, successor_val)
<span class="keyword">return</span> node
</code></pre>
<h3>Full BST Class -- JavaScript</h3>
<pre><code><span class="lang-label">JavaScript</span>
<span class="keyword">class</span> <span class="function">BST</span> {
<span class="function">constructor</span>() {
<span class="keyword">this</span>.root = <span class="keyword">null</span>;
}
<span class="comment">// ---- INSERT ----</span>
<span class="function">insert</span>(val) {
<span class="keyword">this</span>.root = <span class="keyword">this</span>._insert(<span class="keyword">this</span>.root, val);
}
<span class="function">_insert</span>(node, val) {
<span class="keyword">if</span> (!node) <span class="keyword">return new</span> <span class="function">TreeNode</span>(val);
<span class="keyword">if</span> (val < node.val) node.left = <span class="keyword">this</span>._insert(node.left, val);
<span class="keyword">else if</span> (val > node.val) node.right = <span class="keyword">this</span>._insert(node.right, val);
<span class="keyword">return</span> node;
}
<span class="comment">// ---- SEARCH ----</span>
<span class="function">search</span>(val) {
<span class="keyword">return</span> <span class="keyword">this</span>._search(<span class="keyword">this</span>.root, val);
}
<span class="function">_search</span>(node, val) {
<span class="keyword">if</span> (!node) <span class="keyword">return false</span>;
<span class="keyword">if</span> (val === node.val) <span class="keyword">return true</span>;
<span class="keyword">if</span> (val < node.val) <span class="keyword">return</span> <span class="keyword">this</span>._search(node.left, val);
<span class="keyword">return</span> <span class="keyword">this</span>._search(node.right, val);
}
<span class="comment">// ---- FIND MIN / MAX ----</span>
<span class="function">findMin</span>(node = <span class="keyword">this</span>.root) {
<span class="keyword">while</span> (node.left) node = node.left;
<span class="keyword">return</span> node.val;
}
<span class="function">findMax</span>(node = <span class="keyword">this</span>.root) {
<span class="keyword">while</span> (node.right) node = node.right;
<span class="keyword">return</span> node.val;
}
<span class="comment">// ---- DELETE ----</span>
<span class="function">delete</span>(val) {
<span class="keyword">this</span>.root = <span class="keyword">this</span>._delete(<span class="keyword">this</span>.root, val);
}
<span class="function">_delete</span>(node, val) {
<span class="keyword">if</span> (!node) <span class="keyword">return null</span>;
<span class="keyword">if</span> (val < node.val) {
node.left = <span class="keyword">this</span>._delete(node.left, val);
} <span class="keyword">else if</span> (val > node.val) {
node.right = <span class="keyword">this</span>._delete(node.right, val);
} <span class="keyword">else</span> {
<span class="comment">// Case 1: Leaf node</span>
<span class="keyword">if</span> (!node.left && !node.right) <span class="keyword">return null</span>;
<span class="comment">// Case 2: One child</span>
<span class="keyword">if</span> (!node.left) <span class="keyword">return</span> node.right;
<span class="keyword">if</span> (!node.right) <span class="keyword">return</span> node.left;
<span class="comment">// Case 3: Two children</span>
<span class="keyword">const</span> successorVal = <span class="keyword">this</span>.<span class="function">findMin</span>(node.right);
node.val = successorVal;
node.right = <span class="keyword">this</span>._delete(node.right, successorVal);
}
<span class="keyword">return</span> node;
}
}
</code></pre>
<h3>Understanding Delete -- The Three Cases</h3>
<div class="example-box">
<div class="label">Case 1: Deleting a Leaf Node</div>
<p>Simply remove it. No children to worry about.</p>
<pre><code> Delete 4:
<span class="number">5</span> <span class="number">5</span>
/ \ --> / \
<span class="number">3</span> <span class="number">7</span> <span class="number">3</span> <span class="number">7</span>
/
<span class="number">4</span> (removed)</code></pre>
</div>
<div class="example-box">
<div class="label">Case 2: Node with One Child</div>
<p>Replace the node with its only child.</p>
<pre><code> Delete 3 (has left child only):
<span class="number">5</span> <span class="number">5</span>
/ \ --> / \
<span class="number">3</span> <span class="number">7</span> <span class="number">2</span> <span class="number">7</span>
/
<span class="number">2</span></code></pre>
</div>
<div class="example-box">
<div class="label">Case 3: Node with Two Children</div>
<p>Find the <strong>inorder successor</strong> (smallest value in the right subtree), copy its value to the current node, then delete the successor.</p>
<pre><code> Delete 5 (root, has two children):
<span class="number">5</span> Step 1: Find inorder <span class="number">6</span>
/ \ successor = 6 / \
<span class="number">3</span> <span class="number">8</span> Step 2: Copy 6 to root <span class="number">3</span> <span class="number">8</span>
/ \ Step 3: Delete 6 \
<span class="number">6</span> <span class="number">9</span> from right subtree <span class="number">9</span>
\
<span class="number">7</span></code></pre>
</div>
</section>
<!-- ============================================================ -->
<!-- SECTION 4: TREE TRAVERSALS -->
<!-- ============================================================ -->
<section id="traversals">
<h2>4. Tree Traversals</h2>
<p>Tree traversal means visiting every node exactly once. There are two main approaches: <strong>Depth-First Search (DFS)</strong> and <strong>Breadth-First Search (BFS)</strong>.</p>
<div class="formula-box">
Reference tree for all traversal examples:<br><br>
4<br>
/ \<br>
2 6<br>
/ \ / \<br>
1 3 5 7
</div>
<div class="formula-box">
<strong>Traversal Use-Case Axioms:</strong><br><br>
• <strong>Inorder (Left, Root, Right):</strong> BST → produces sorted output<br>
• <strong>Preorder (Root, Left, Right):</strong> Copy/serialize tree, prefix expressions<br>
• <strong>Postorder (Left, Right, Root):</strong> Delete tree, compute subtree values bottom-up<br>
• <strong>Level-order (BFS):</strong> Shortest path in unweighted tree, level-by-level processing<br><br>
<strong>Recursive template:</strong> return_value = combine(recurse(left), recurse(right), node.val)
</div>
<!-- INORDER -->
<h3>Inorder Traversal (Left, Root, Right)</h3>
<p>Visit left subtree, then current node, then right subtree. For a BST, this gives values in <strong>sorted ascending order</strong>.</p>
<pre><code>
Inorder walk-through:
<span class="number">4</span>
/ \
<span class="number">2</span> <span class="number">6</span>
/ \ / \
<span class="number">1</span> <span class="number">3</span> <span class="number">5</span> <span class="number">7</span>
Visit order: 1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 7 (sorted!)
</code></pre>
<pre><code><span class="lang-label">Python</span>
<span class="comment"># Recursive Inorder</span>
<span class="keyword">def</span> <span class="function">inorder_recursive</span>(root):
result = []
<span class="keyword">def</span> <span class="function">dfs</span>(node):
<span class="keyword">if not</span> node:
<span class="keyword">return</span>
dfs(node.left) <span class="comment"># Left</span>
result.append(node.val) <span class="comment"># Root</span>
dfs(node.right) <span class="comment"># Right</span>
dfs(root)
<span class="keyword">return</span> result
<span class="comment"># Iterative Inorder (using stack)</span>
<span class="keyword">def</span> <span class="function">inorder_iterative</span>(root):
result = []
stack = []
current = root
<span class="keyword">while</span> current <span class="keyword">or</span> stack:
<span class="comment"># Go as far left as possible</span>
<span class="keyword">while</span> current:
stack.append(current)
current = current.left
<span class="comment"># Process node</span>
current = stack.pop()
result.append(current.val)
<span class="comment"># Move to right subtree</span>
current = current.right
<span class="keyword">return</span> result
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="comment">// Recursive Inorder</span>
<span class="keyword">function</span> <span class="function">inorderRecursive</span>(root) {
<span class="keyword">const</span> result = [];
<span class="keyword">function</span> <span class="function">dfs</span>(node) {
<span class="keyword">if</span> (!node) <span class="keyword">return</span>;
dfs(node.left); <span class="comment">// Left</span>
result.push(node.val); <span class="comment">// Root</span>
dfs(node.right); <span class="comment">// Right</span>
}
dfs(root);
<span class="keyword">return</span> result;
}
<span class="comment">// Iterative Inorder (using stack)</span>
<span class="keyword">function</span> <span class="function">inorderIterative</span>(root) {
<span class="keyword">const</span> result = [];
<span class="keyword">const</span> stack = [];
<span class="keyword">let</span> current = root;
<span class="keyword">while</span> (current || stack.length) {
<span class="keyword">while</span> (current) {
stack.push(current);
current = current.left;
}
current = stack.pop();
result.push(current.val);
current = current.right;
}
<span class="keyword">return</span> result;
}
</code></pre>
<!-- PREORDER -->
<h3>Preorder Traversal (Root, Left, Right)</h3>
<p>Visit current node first, then left subtree, then right subtree. Useful for <strong>copying/serializing</strong> a tree.</p>
<pre><code>
Preorder walk-through:
<span class="number">4</span>
/ \
<span class="number">2</span> <span class="number">6</span>
/ \ / \
<span class="number">1</span> <span class="number">3</span> <span class="number">5</span> <span class="number">7</span>
Visit order: 4 -> 2 -> 1 -> 3 -> 6 -> 5 -> 7
</code></pre>
<pre><code><span class="lang-label">Python</span>
<span class="comment"># Recursive Preorder</span>
<span class="keyword">def</span> <span class="function">preorder_recursive</span>(root):
result = []
<span class="keyword">def</span> <span class="function">dfs</span>(node):
<span class="keyword">if not</span> node:
<span class="keyword">return</span>
result.append(node.val) <span class="comment"># Root</span>
dfs(node.left) <span class="comment"># Left</span>
dfs(node.right) <span class="comment"># Right</span>
dfs(root)
<span class="keyword">return</span> result
<span class="comment"># Iterative Preorder (using stack)</span>
<span class="keyword">def</span> <span class="function">preorder_iterative</span>(root):
<span class="keyword">if not</span> root:
<span class="keyword">return</span> []
result = []
stack = [root]
<span class="keyword">while</span> stack:
node = stack.pop()
result.append(node.val)
<span class="comment"># Push right first so left is processed first</span>
<span class="keyword">if</span> node.right:
stack.append(node.right)
<span class="keyword">if</span> node.left:
stack.append(node.left)
<span class="keyword">return</span> result
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="comment">// Recursive Preorder</span>
<span class="keyword">function</span> <span class="function">preorderRecursive</span>(root) {
<span class="keyword">const</span> result = [];
<span class="keyword">function</span> <span class="function">dfs</span>(node) {
<span class="keyword">if</span> (!node) <span class="keyword">return</span>;
result.push(node.val); <span class="comment">// Root</span>
dfs(node.left); <span class="comment">// Left</span>
dfs(node.right); <span class="comment">// Right</span>
}
dfs(root);
<span class="keyword">return</span> result;
}
<span class="comment">// Iterative Preorder (using stack)</span>
<span class="keyword">function</span> <span class="function">preorderIterative</span>(root) {
<span class="keyword">if</span> (!root) <span class="keyword">return</span> [];
<span class="keyword">const</span> result = [];
<span class="keyword">const</span> stack = [root];
<span class="keyword">while</span> (stack.length) {
<span class="keyword">const</span> node = stack.pop();
result.push(node.val);
<span class="keyword">if</span> (node.right) stack.push(node.right);
<span class="keyword">if</span> (node.left) stack.push(node.left);
}
<span class="keyword">return</span> result;
}
</code></pre>
<!-- POSTORDER -->
<h3>Postorder Traversal (Left, Right, Root)</h3>
<p>Visit left subtree, then right subtree, then current node. Useful for <strong>deletion</strong> (delete children before parent) and evaluating expression trees.</p>
<pre><code>
Postorder walk-through:
<span class="number">4</span>
/ \
<span class="number">2</span> <span class="number">6</span>
/ \ / \
<span class="number">1</span> <span class="number">3</span> <span class="number">5</span> <span class="number">7</span>
Visit order: 1 -> 3 -> 2 -> 5 -> 7 -> 6 -> 4
</code></pre>
<pre><code><span class="lang-label">Python</span>
<span class="comment"># Recursive Postorder</span>
<span class="keyword">def</span> <span class="function">postorder_recursive</span>(root):
result = []
<span class="keyword">def</span> <span class="function">dfs</span>(node):
<span class="keyword">if not</span> node:
<span class="keyword">return</span>
dfs(node.left) <span class="comment"># Left</span>
dfs(node.right) <span class="comment"># Right</span>
result.append(node.val) <span class="comment"># Root</span>
dfs(root)
<span class="keyword">return</span> result
<span class="comment"># Iterative Postorder (using two stacks)</span>
<span class="keyword">def</span> <span class="function">postorder_iterative</span>(root):
<span class="keyword">if not</span> root:
<span class="keyword">return</span> []
result = []
stack = [root]
<span class="keyword">while</span> stack:
node = stack.pop()
result.append(node.val)
<span class="keyword">if</span> node.left:
stack.append(node.left)
<span class="keyword">if</span> node.right:
stack.append(node.right)
<span class="keyword">return</span> result[::-<span class="number">1</span>] <span class="comment"># Reverse the result</span>
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="comment">// Recursive Postorder</span>
<span class="keyword">function</span> <span class="function">postorderRecursive</span>(root) {
<span class="keyword">const</span> result = [];
<span class="keyword">function</span> <span class="function">dfs</span>(node) {
<span class="keyword">if</span> (!node) <span class="keyword">return</span>;
dfs(node.left); <span class="comment">// Left</span>
dfs(node.right); <span class="comment">// Right</span>
result.push(node.val); <span class="comment">// Root</span>
}
dfs(root);
<span class="keyword">return</span> result;
}
<span class="comment">// Iterative Postorder</span>
<span class="keyword">function</span> <span class="function">postorderIterative</span>(root) {
<span class="keyword">if</span> (!root) <span class="keyword">return</span> [];
<span class="keyword">const</span> result = [];
<span class="keyword">const</span> stack = [root];
<span class="keyword">while</span> (stack.length) {
<span class="keyword">const</span> node = stack.pop();
result.push(node.val);
<span class="keyword">if</span> (node.left) stack.push(node.left);
<span class="keyword">if</span> (node.right) stack.push(node.right);
}
<span class="keyword">return</span> result.reverse();
}
</code></pre>
<!-- BFS / LEVEL ORDER -->
<h3>Level Order Traversal (BFS)</h3>
<p>Visit nodes level by level, from left to right. Uses a <strong>queue</strong> instead of a stack.</p>
<pre><code>
Level order walk-through:
<span class="number">4</span> Level 0: [4]
/ \
<span class="number">2</span> <span class="number">6</span> Level 1: [2, 6]
/ \ / \
<span class="number">1</span> <span class="number">3</span> <span class="number">5</span> <span class="number">7</span> Level 2: [1, 3, 5, 7]
Result: [[4], [2, 6], [1, 3, 5, 7]]
</code></pre>
<pre><code><span class="lang-label">Python</span>
<span class="keyword">from</span> collections <span class="keyword">import</span> deque
<span class="keyword">def</span> <span class="function">level_order</span>(root):
<span class="keyword">if not</span> root:
<span class="keyword">return</span> []
result = []
queue = deque([root])
<span class="keyword">while</span> queue:
level_size = <span class="builtin">len</span>(queue)
current_level = []
<span class="keyword">for</span> _ <span class="keyword">in</span> <span class="builtin">range</span>(level_size):
node = queue.popleft()
current_level.append(node.val)
<span class="keyword">if</span> node.left:
queue.append(node.left)
<span class="keyword">if</span> node.right:
queue.append(node.right)
result.append(current_level)
<span class="keyword">return</span> result
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="keyword">function</span> <span class="function">levelOrder</span>(root) {
<span class="keyword">if</span> (!root) <span class="keyword">return</span> [];
<span class="keyword">const</span> result = [];
<span class="keyword">const</span> queue = [root];
<span class="keyword">while</span> (queue.length) {
<span class="keyword">const</span> levelSize = queue.length;
<span class="keyword">const</span> currentLevel = [];
<span class="keyword">for</span> (<span class="keyword">let</span> i = <span class="number">0</span>; i < levelSize; i++) {
<span class="keyword">const</span> node = queue.shift();
currentLevel.push(node.val);
<span class="keyword">if</span> (node.left) queue.push(node.left);
<span class="keyword">if</span> (node.right) queue.push(node.right);
}
result.push(currentLevel);
}
<span class="keyword">return</span> result;
}
</code></pre>
<div class="tip-box">
<div class="label">Traversal Cheat Sheet</div>
<p><strong>Inorder</strong> (L, Root, R) -- sorted order for BST, most common</p>
<p><strong>Preorder</strong> (Root, L, R) -- copy/serialize tree, build tree from traversal</p>
<p><strong>Postorder</strong> (L, R, Root) -- delete tree, evaluate expressions</p>
<p><strong>Level Order</strong> (BFS) -- shortest path, level-by-level processing</p>
</div>
</section>
<!-- ============================================================ -->
<!-- SECTION 5: COMMON TREE PROBLEMS -->
<!-- ============================================================ -->
<section id="common-problems">
<h2>5. Common Tree Problems & Patterns</h2>
<!-- MAX DEPTH -->
<h3>Maximum Depth of Binary Tree</h3>
<p>Find the maximum depth (number of nodes along the longest path from root to the farthest leaf). This is the classic recursive tree problem.</p>
<div class="formula-box">
maxDepth(node) = 1 + max(maxDepth(left), maxDepth(right))<br>
Base case: maxDepth(null) = 0
</div>
<pre><code><span class="lang-label">Python</span>
<span class="keyword">def</span> <span class="function">max_depth</span>(root):
<span class="keyword">if not</span> root:
<span class="keyword">return</span> <span class="number">0</span>
left_depth = max_depth(root.left)
right_depth = max_depth(root.right)
<span class="keyword">return</span> <span class="number">1</span> + <span class="builtin">max</span>(left_depth, right_depth)
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="keyword">function</span> <span class="function">maxDepth</span>(root) {
<span class="keyword">if</span> (!root) <span class="keyword">return</span> <span class="number">0</span>;
<span class="keyword">const</span> leftDepth = maxDepth(root.left);
<span class="keyword">const</span> rightDepth = maxDepth(root.right);
<span class="keyword">return</span> <span class="number">1</span> + Math.max(leftDepth, rightDepth);
}
</code></pre>
<pre><code>
Walk-through:
<span class="number">3</span> maxDepth(3) = 1 + max(2, 1) = 3
/ \
<span class="number">9</span> <span class="number">20</span> maxDepth(9) = 1 + max(0, 0) = 1 | maxDepth(20) = 1 + max(1, 1) = 2
/ \
<span class="number">15</span> <span class="number">7</span> maxDepth(15) = 1 | maxDepth(7) = 1
Answer: 3
</code></pre>
<!-- VALIDATE BST -->
<h3>Validate Binary Search Tree</h3>
<p>Check if a binary tree is a valid BST. The trick: each node must be within a valid <strong>range</strong> (min, max), not just compared to its immediate parent.</p>
<div class="warning-box">
<div class="label">Common Mistake</div>
<p>Do NOT just check <code>node.left.val < node.val < node.right.val</code>. This misses cases where a deeper node violates the BST property with an ancestor.</p>
<pre><code> <span class="number">5</span>
/ \
<span class="number">1</span> <span class="number">7</span>
/ \
<span class="number">3</span> <span class="number">8</span> <-- 3 is less than 5, violates BST!
(3 < 7 passes local check but 3 < 5 fails)</code></pre>
</div>
<pre><code><span class="lang-label">Python</span>
<span class="comment"># Approach 1: Min/Max Range</span>
<span class="keyword">def</span> <span class="function">is_valid_bst</span>(root):
<span class="keyword">def</span> <span class="function">validate</span>(node, low=<span class="builtin">float</span>(<span class="string">'-inf'</span>), high=<span class="builtin">float</span>(<span class="string">'inf'</span>)):
<span class="keyword">if not</span> node:
<span class="keyword">return</span> <span class="keyword">True</span>
<span class="keyword">if</span> node.val <= low <span class="keyword">or</span> node.val >= high:
<span class="keyword">return</span> <span class="keyword">False</span>
<span class="keyword">return</span> (validate(node.left, low, node.val) <span class="keyword">and</span>
validate(node.right, node.val, high))
<span class="keyword">return</span> validate(root)
<span class="comment"># Approach 2: Inorder traversal should be sorted</span>
<span class="keyword">def</span> <span class="function">is_valid_bst_inorder</span>(root):
prev = [<span class="builtin">float</span>(<span class="string">'-inf'</span>)]
<span class="keyword">def</span> <span class="function">inorder</span>(node):
<span class="keyword">if not</span> node:
<span class="keyword">return</span> <span class="keyword">True</span>
<span class="keyword">if not</span> inorder(node.left):
<span class="keyword">return</span> <span class="keyword">False</span>
<span class="keyword">if</span> node.val <= prev[<span class="number">0</span>]:
<span class="keyword">return</span> <span class="keyword">False</span>
prev[<span class="number">0</span>] = node.val
<span class="keyword">return</span> inorder(node.right)
<span class="keyword">return</span> inorder(root)
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="comment">// Approach 1: Min/Max Range</span>
<span class="keyword">function</span> <span class="function">isValidBST</span>(root) {
<span class="keyword">function</span> <span class="function">validate</span>(node, low = -<span class="builtin">Infinity</span>, high = <span class="builtin">Infinity</span>) {
<span class="keyword">if</span> (!node) <span class="keyword">return true</span>;
<span class="keyword">if</span> (node.val <= low || node.val >= high) <span class="keyword">return false</span>;
<span class="keyword">return</span> validate(node.left, low, node.val) &&
validate(node.right, node.val, high);
}
<span class="keyword">return</span> validate(root);
}
<span class="comment">// Approach 2: Inorder traversal should be sorted</span>
<span class="keyword">function</span> <span class="function">isValidBSTInorder</span>(root) {
<span class="keyword">let</span> prev = -<span class="builtin">Infinity</span>;
<span class="keyword">function</span> <span class="function">inorder</span>(node) {
<span class="keyword">if</span> (!node) <span class="keyword">return true</span>;
<span class="keyword">if</span> (!inorder(node.left)) <span class="keyword">return false</span>;
<span class="keyword">if</span> (node.val <= prev) <span class="keyword">return false</span>;
prev = node.val;
<span class="keyword">return</span> inorder(node.right);
}
<span class="keyword">return</span> inorder(root);
}
</code></pre>
<!-- INVERT BINARY TREE -->
<h3>Invert Binary Tree</h3>
<p>Swap left and right children at every node. The famous question that Max Howell (Homebrew creator) couldn't solve in a Google interview.</p>
<pre><code>
Before: After:
<span class="number">4</span> <span class="number">4</span>
/ \ / \
<span class="number">2</span> <span class="number">7</span> <span class="number">7</span> <span class="number">2</span>
/ \ / \ / \ / \
<span class="number">1</span> <span class="number">3</span> <span class="number">6</span> <span class="number">9</span> <span class="number">9</span> <span class="number">6</span> <span class="number">3</span> <span class="number">1</span>
</code></pre>
<pre><code><span class="lang-label">Python</span>
<span class="keyword">def</span> <span class="function">invert_tree</span>(root):
<span class="keyword">if not</span> root:
<span class="keyword">return</span> <span class="keyword">None</span>
<span class="comment"># Swap left and right</span>
root.left, root.right = root.right, root.left
<span class="comment"># Recursively invert subtrees</span>
invert_tree(root.left)
invert_tree(root.right)
<span class="keyword">return</span> root
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="keyword">function</span> <span class="function">invertTree</span>(root) {
<span class="keyword">if</span> (!root) <span class="keyword">return null</span>;
<span class="comment">// Swap left and right</span>
[root.left, root.right] = [root.right, root.left];
<span class="comment">// Recursively invert subtrees</span>
invertTree(root.left);
invertTree(root.right);
<span class="keyword">return</span> root;
}
</code></pre>
<!-- LOWEST COMMON ANCESTOR -->
<h3>Lowest Common Ancestor of a BST</h3>
<p>Find the lowest node that is an ancestor of both p and q. In a BST, we can leverage the ordering property.</p>
<div class="formula-box">
BST LCA Logic:<br>
- If both p and q are smaller than root -> LCA is in left subtree<br>
- If both p and q are larger than root -> LCA is in right subtree<br>
- Otherwise, root IS the LCA (split point)
</div>
<pre><code><span class="lang-label">Python</span>
<span class="keyword">def</span> <span class="function">lowest_common_ancestor</span>(root, p, q):
<span class="keyword">while</span> root:
<span class="keyword">if</span> p.val < root.val <span class="keyword">and</span> q.val < root.val:
root = root.left <span class="comment"># Both in left subtree</span>
<span class="keyword">elif</span> p.val > root.val <span class="keyword">and</span> q.val > root.val:
root = root.right <span class="comment"># Both in right subtree</span>
<span class="keyword">else</span>:
<span class="keyword">return</span> root <span class="comment"># Split point = LCA</span>
</code></pre>
<pre><code><span class="lang-label">JavaScript</span>
<span class="keyword">function</span> <span class="function">lowestCommonAncestor</span>(root, p, q) {
<span class="keyword">while</span> (root) {
<span class="keyword">if</span> (p.val < root.val && q.val < root.val) {
root = root.left; <span class="comment">// Both in left subtree</span>
} <span class="keyword">else if</span> (p.val > root.val && q.val > root.val) {
root = root.right; <span class="comment">// Both in right subtree</span>
} <span class="keyword">else</span> {
<span class="keyword">return</span> root; <span class="comment">// Split point = LCA</span>
}
}
}
</code></pre>
<pre><code>
Example: Find LCA of 2 and 8