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<!DOCTYPE html>

<head>

<title>Perfect Binary Tree</title>

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<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

(I)</a></li>

<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

(II)</a></li>

<li><a href="/dsa.html#tree-1" title="Tree based DSA (I)">Tree based DSA (I)</a></li>

<li><a href="/dsa.html#tree-2" title="Tree based DSA (II)">Tree based DSA (II)</a></li>

<li><a href="/dsa.html#graph" title="Graph Data Structures and Algorithm">Graph based

DSA</a></li>

<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

Searching</a></li>

<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

</li>

<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

Programming</a></li>

<li><a href="/dsa.html#other-algorithms" title="Other Algorithms">Other Algorithms</a>

</li>

</ul>

</div>

<h1>Perfect Binary Tree</h1>

<p class="editor-contents__short-description">In this tutorial, you will learn about the perfect binary tree. Also, you will find working examples for checking a perfect binary tree in C, C++, Java and Python.</p>

<p id="definition">A perfect binary tree is a type of binary tree in which every internal node has exactly two child nodes and all the leaf nodes are at the same level.</p>

<figure><img alt="Perfect Binary Tree" src="//cdn.programiz.com/sites/tutorial2program/files/perfect-binary-tree_0.png" title="Perfect Binary Tree" width="288" height="216"><figcaption>Perfect Binary Tree</figcaption></figure><p>All the internal nodes have a degree of 2.</p>

<p>Recursively, a perfect binary tree can be defined as:</p>

<ol><li>If a single node has no children, it is a perfect binary tree of height <code>h = 0</code>,</li>

<li>If a node has <code>h > 0</code>, it is a perfect binary tree if both of its subtrees are of height <code>h - 1</code> and are non-overlapping.</li>

</ol><figure><img alt="Perfect Binary Tree (Recursive Representation)" src="//cdn.programiz.com/sites/tutorial2program/files/perfect-binary-tree-rec.png" title="Perfect Binary Tree" width="555" height="269"><figcaption>Perfect Binary Tree (Recursive Representation)</figcaption></figure><hr><h2 id="code">Python, Java and C/C++ Examples</h2>

<p></p>

<div class="clearfix"></div><p>The following code is for checking whether a tree is a perfect binary tree.</p>

<div id="py1" onclick="changepy()" class="tabbed-editor__node tabbed-editor__node--active"><a href="#python-code">Python</a></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="python hljs"><span class="hljs-comment"># Checking if a binary tree is a perfect binary tree in Python</span>

<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">newNode</span>:</span>

self.key = k

self.right = self.left = <span class="hljs-literal">None</span>

<span class="hljs-comment"># Calculate the depth</span>

<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">calculateDepth</span><span class="hljs-params">(node)</span>:</span>

d = <span class="hljs-number">0</span>

<span class="hljs-keyword">while</span> (node <span class="hljs-keyword">is</span> <span class="hljs-keyword">not</span> <span class="hljs-literal">None</span>):

d += <span class="hljs-number">1</span>

node = node.left

<span class="hljs-keyword">return</span> d

<span class="hljs-comment"># Check if the tree is perfect binary tree</span>

<span class="hljs-function"><span class="hljs-keyword">def</span> <span class="hljs-title">is_perfect</span><span class="hljs-params">(root, d, level=<span class="hljs-number">0</span>)</span>:</span>

<span class="hljs-comment"># Check if the tree is empty</span>

<span class="hljs-keyword">if</span> (root <span class="hljs-keyword">is</span> <span class="hljs-literal">None</span>):

<span class="hljs-keyword">return</span> <span class="hljs-literal">True</span>

<span class="hljs-comment"># Check the presence of trees</span>

<span class="hljs-keyword">if</span> (root.left <span class="hljs-keyword">is</span> <span class="hljs-literal">None</span> <span class="hljs-keyword">and</span> root.right <span class="hljs-keyword">is</span> <span class="hljs-literal">None</span>):

<span class="hljs-keyword">return</span> (d == level + <span class="hljs-number">1</span>)

<span class="hljs-keyword">if</span> (root.left <span class="hljs-keyword">is</span> <span class="hljs-literal">None</span> <span class="hljs-keyword">or</span> root.right <span class="hljs-keyword">is</span> <span class="hljs-literal">None</span>):

<span class="hljs-keyword">return</span> <span class="hljs-literal">False</span>

<span class="hljs-keyword">return</span> (is_perfect(root.left, d, level + <span class="hljs-number">1</span>) <span class="hljs-keyword">and</span>

is_perfect(root.right, d, level + <span class="hljs-number">1</span>))

root = <span class="hljs-literal">None</span>

root = newNode(<span class="hljs-number">1</span>)

root.left = newNode(<span class="hljs-number">2</span>)

root.right = newNode(<span class="hljs-number">3</span>)

root.left.left = newNode(<span class="hljs-number">4</span>)

root.left.right = newNode(<span class="hljs-number">5</span>)

<span class="hljs-keyword">if</span> (is_perfect(root, calculateDepth(root))):

<span class="hljs-keyword">print</span>(<span class="hljs-string">"The tree is a perfect binary tree"</span>)

<span class="hljs-keyword">else</span>:

<span class="hljs-keyword">print</span>(<span class="hljs-string">"The tree is not a perfect binary tree"</span>)</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="java hljs"><span class="hljs-comment">// Checking if a binary tree is a perfect binary tree in Java</span>

<span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">PerfectBinaryTree</span> </span>{

<span class="hljs-keyword">static</span> <span class="hljs-class"><span class="hljs-keyword">class</span> <span class="hljs-title">Node</span> </span>{

<span class="hljs-keyword">int</span> key;

Node left, right;

}

<span class="hljs-comment">// Calculate the depth</span>

<span class="hljs-function"><span class="hljs-keyword">static</span> <span class="hljs-keyword">int</span> <span class="hljs-title">depth</span><span class="hljs-params">(Node node)</span> </span>{

<span class="hljs-keyword">int</span> d = <span class="hljs-number">0</span>;

<span class="hljs-keyword">while</span> (node != <span class="hljs-keyword">null</span>) {

d++;

node = node.left;

}

<span class="hljs-keyword">return</span> d;

}

<span class="hljs-comment">// Check if the tree is perfect binary tree</span>

<span class="hljs-function"><span class="hljs-keyword">static</span> <span class="hljs-keyword">boolean</span> <span class="hljs-title">is_perfect</span><span class="hljs-params">(Node root, <span class="hljs-keyword">int</span> d, <span class="hljs-keyword">int</span> level)</span> </span>{

<span class="hljs-comment">// Check if the tree is empty</span>

<span class="hljs-keyword">if</span> (root == <span class="hljs-keyword">null</span>)

<span class="hljs-keyword">return</span> <span class="hljs-keyword">true</span>;

<span class="hljs-comment">// If for children</span>

<span class="hljs-keyword">if</span> (root.left == <span class="hljs-keyword">null</span> && root.right == <span class="hljs-keyword">null</span>)

<span class="hljs-keyword">return</span> (d == level + <span class="hljs-number">1</span>);

<span class="hljs-keyword">if</span> (root.left == <span class="hljs-keyword">null</span> || root.right == <span class="hljs-keyword">null</span>)

<span class="hljs-keyword">return</span> <span class="hljs-keyword">false</span>;

<span class="hljs-keyword">return</span> is_perfect(root.left, d, level + <span class="hljs-number">1</span>) && is_perfect(root.right, d, level + <span class="hljs-number">1</span>);

}

<span class="hljs-comment">// Wrapper function</span>

<span class="hljs-function"><span class="hljs-keyword">static</span> <span class="hljs-keyword">boolean</span> <span class="hljs-title">is_Perfect</span><span class="hljs-params">(Node root)</span> </span>{

<span class="hljs-keyword">int</span> d = depth(root);

<span class="hljs-keyword">return</span> is_perfect(root, d, <span class="hljs-number">0</span>);

}

<span class="hljs-comment">// Create a new node</span>

<span class="hljs-function"><span class="hljs-keyword">static</span> Node <span class="hljs-title">newNode</span><span class="hljs-params">(<span class="hljs-keyword">int</span> k)</span> </span>{

Node node = <span class="hljs-keyword">new</span> Node();

node.key = k;

node.right = <span class="hljs-keyword">null</span>;

node.left = <span class="hljs-keyword">null</span>;

<span class="hljs-keyword">return</span> node;

}

<span class="hljs-function"><span class="hljs-keyword">public</span> <span class="hljs-keyword">static</span> <span class="hljs-keyword">void</span> <span class="hljs-title">main</span><span class="hljs-params">(String args[])</span> </span>{

Node root = <span class="hljs-keyword">null</span>;

root = newNode(<span class="hljs-number">1</span>);

root.left = newNode(<span class="hljs-number">2</span>);

root.right = newNode(<span class="hljs-number">3</span>);

root.left.left = newNode(<span class="hljs-number">4</span>);

root.left.right = newNode(<span class="hljs-number">5</span>);

<span class="hljs-keyword">if</span> (is_Perfect(root) == <span class="hljs-keyword">true</span>)

System.out.println(<span class="hljs-string">"The tree is a perfect binary tree"</span>);

System.out.println(<span class="hljs-string">"The tree is not a perfect binary tree"</span>);

}

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="c hljs cpp"><span class="hljs-comment">// Checking if a binary tree is a perfect binary tree in C</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdbool.h></span></span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdio.h></span></span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><stdlib.h></span></span>

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">node</span> {</span>

<span class="hljs-keyword">int</span> data;

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">node</span> *<span class="hljs-title">left</span>;</span>

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">node</span> *<span class="hljs-title">right</span>;</span>

};

<span class="hljs-comment">// Creating a new node</span>

<span class="hljs-function">struct node *<span class="hljs-title">newnode</span><span class="hljs-params">(<span class="hljs-keyword">int</span> data)</span> </span>{

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">node</span> *<span class="hljs-title">node</span> = (<span class="hljs-title">struct</span> <span class="hljs-title">node</span> *)<span class="hljs-title">malloc</span>(<span class="hljs-title">sizeof</span>(<span class="hljs-title">struct</span> <span class="hljs-title">node</span>));</span>

node->data = data;

node->left = <span class="hljs-literal">NULL</span>;

node->right = <span class="hljs-literal">NULL</span>;

<span class="hljs-keyword">return</span> (node);

}

<span class="hljs-comment">// Calculate the depth</span>

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">depth</span><span class="hljs-params">(struct node *node)</span> </span>{

<span class="hljs-keyword">int</span> d = <span class="hljs-number">0</span>;

<span class="hljs-keyword">while</span> (node != <span class="hljs-literal">NULL</span>) {

d++;

node = node->left;

}

<span class="hljs-keyword">return</span> d;

}

<span class="hljs-comment">// Check if the tree is perfect</span>

<span class="hljs-function"><span class="hljs-keyword">bool</span> <span class="hljs-title">is_perfect</span><span class="hljs-params">(struct node *root, <span class="hljs-keyword">int</span> d, <span class="hljs-keyword">int</span> level)</span> </span>{

<span class="hljs-comment">// Check if the tree is empty</span>

<span class="hljs-keyword">if</span> (root == <span class="hljs-literal">NULL</span>)

<span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;

<span class="hljs-comment">// Check the presence of children</span>

<span class="hljs-keyword">if</span> (root->left == <span class="hljs-literal">NULL</span> && root->right == <span class="hljs-literal">NULL</span>)

<span class="hljs-keyword">return</span> (d == level + <span class="hljs-number">1</span>);

<span class="hljs-keyword">if</span> (root->left == <span class="hljs-literal">NULL</span> || root->right == <span class="hljs-literal">NULL</span>)

<span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;

<span class="hljs-keyword">return</span> is_perfect(root->left, d, level + <span class="hljs-number">1</span>) &&

is_perfect(root->right, d, level + <span class="hljs-number">1</span>);

}

<span class="hljs-comment">// Wrapper function</span>

<span class="hljs-function"><span class="hljs-keyword">bool</span> <span class="hljs-title">is_Perfect</span><span class="hljs-params">(struct node *root)</span> </span>{

<span class="hljs-keyword">int</span> d = depth(root);

<span class="hljs-keyword">return</span> is_perfect(root, d, <span class="hljs-number">0</span>);

}

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">node</span> *<span class="hljs-title">root</span> = <span class="hljs-title">NULL</span>;</span>

root = newnode(<span class="hljs-number">1</span>);

root->left = newnode(<span class="hljs-number">2</span>);

root->right = newnode(<span class="hljs-number">3</span>);

root->left->left = newnode(<span class="hljs-number">4</span>);

root->left->right = newnode(<span class="hljs-number">5</span>);

root->right->left = newnode(<span class="hljs-number">6</span>);

<span class="hljs-keyword">if</span> (is_Perfect(root))

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"The tree is a perfect binary tree\n"</span>);

<span class="hljs-built_in">printf</span>(<span class="hljs-string">"The tree is not a perfect binary tree\n"</span>);

}</code></pre></div>

</div>

<div class="pre-code-wrapper"><div title="Click to copy" class="copy-code-button"></div><pre class="exec" style="max-height: 600px;"><code class="cpp hljs"><span class="hljs-comment">// Checking if a binary tree is a perfect binary tree in C++</span>

<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span> <span class="hljs-meta-string"><iostream></span></span>

<span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> <span class="hljs-built_in">std</span>;

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Node</span> {</span>

<span class="hljs-keyword">int</span> key;

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Node</span> *<span class="hljs-title">left</span>, *<span class="hljs-title">right</span>;</span>

};

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">depth</span><span class="hljs-params">(Node *node)</span> </span>{

<span class="hljs-keyword">int</span> d = <span class="hljs-number">0</span>;

<span class="hljs-keyword">while</span> (node != <span class="hljs-literal">NULL</span>) {

d++;

node = node->left;

}

<span class="hljs-keyword">return</span> d;

}

<span class="hljs-function"><span class="hljs-keyword">bool</span> <span class="hljs-title">isPerfectR</span><span class="hljs-params">(struct Node *root, <span class="hljs-keyword">int</span> d, <span class="hljs-keyword">int</span> level = <span class="hljs-number">0</span>)</span> </span>{

<span class="hljs-keyword">if</span> (root == <span class="hljs-literal">NULL</span>)

<span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;

<span class="hljs-keyword">if</span> (root->left == <span class="hljs-literal">NULL</span> && root->right == <span class="hljs-literal">NULL</span>)

<span class="hljs-keyword">return</span> (d == level + <span class="hljs-number">1</span>);

<span class="hljs-keyword">if</span> (root->left == <span class="hljs-literal">NULL</span> || root->right == <span class="hljs-literal">NULL</span>)

<span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;

<span class="hljs-keyword">return</span> isPerfectR(root->left, d, level + <span class="hljs-number">1</span>) &&

isPerfectR(root->right, d, level + <span class="hljs-number">1</span>);

}

<span class="hljs-function"><span class="hljs-keyword">bool</span> <span class="hljs-title">isPerfect</span><span class="hljs-params">(Node *root)</span> </span>{

<span class="hljs-keyword">int</span> d = depth(root);

<span class="hljs-keyword">return</span> isPerfectR(root, d);

}

<span class="hljs-function">struct Node *<span class="hljs-title">newNode</span><span class="hljs-params">(<span class="hljs-keyword">int</span> k)</span> </span>{

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Node</span> *<span class="hljs-title">node</span> = <span class="hljs-title">new</span> <span class="hljs-title">Node</span>;</span>

node->key = k;

node->right = node->left = <span class="hljs-literal">NULL</span>;

<span class="hljs-keyword">return</span> node;

}

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{

<span class="hljs-class"><span class="hljs-keyword">struct</span> <span class="hljs-title">Node</span> *<span class="hljs-title">root</span> = <span class="hljs-title">NULL</span>;</span>

root = newNode(<span class="hljs-number">1</span>);

root->left = newNode(<span class="hljs-number">2</span>);

root->right = newNode(<span class="hljs-number">3</span>);

root->left->left = newNode(<span class="hljs-number">4</span>);

root->left->right = newNode(<span class="hljs-number">5</span>);

root->right->left = newNode(<span class="hljs-number">6</span>);

<span class="hljs-keyword">if</span> (isPerfect(root))

<span class="hljs-built_in">cout</span> << <span class="hljs-string">"The tree is a perfect binary tree\n"</span>;

<span class="hljs-built_in">cout</span> << <span class="hljs-string">"The tree is not a perfect binary tree\n"</span>;

}</code></pre></div>

</div>

<hr><h2 id="theorem">Perfect Binary Tree Theorems</h2>

<ol><li>A perfect binary tree of height h has <code>2<sup>h + 1</sup> – 1</code> node.</li>

<li>A perfect binary tree with n nodes has height <code>log(n + 1) – 1 = Θ(ln(n))</code>.</li>

<li>A perfect binary tree of height h has <code>2<sup>h</sup></code> leaf nodes.</li>

<li>The average depth of a node in a perfect binary tree is <code>Θ(ln(n))</code>.</li>

</ol></div>

</div>

<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#code">Python, Java and C/C++ Examples</a></li>

<li><a href="#theorem">Perfect Binary Tree Theorems</a></li>

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