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<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></nav> <ul class="sidebar-links"><li><section class="sidebar-group depth-0"><p class="sidebar-heading open"><span>Algorithm</span> <!----></p> <ul class="sidebar-links sidebar-group-items"><li><a href="/algorithm/" aria-current="page" class="sidebar-link">Disclaimer</a></li><li><a href="/algorithm/getting-started-with-algorithm.html" class="sidebar-link">Getting started with algorithm</a></li><li><a href="/algorithm/algorithm-complexity.html" 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href="/algorithm/binary-tree-traversals.html" class="sidebar-link">Binary Tree traversals</a></li><li><a href="/algorithm/lowest-common-ancestor-of-a-binary-tree.html" class="sidebar-link">Lowest common ancestor of a Binary Tree</a></li><li><a href="/algorithm/graph.html" class="sidebar-link">Graph</a></li><li><a href="/algorithm/graph-traversals.html" class="sidebar-link">Graph Traversals</a></li><li><a href="/algorithm/dijkstras-algorithm.html" class="sidebar-link">Dijkstra’s Algorithm</a></li><li><a href="/algorithm/a-pathfinding.html" class="sidebar-link">A* Pathfinding</a></li><li><a href="/algorithm/a-pathfinding-algorithm.html" class="sidebar-link">A* Pathfinding Algorithm</a></li><li><a href="/algorithm/dynamic-programming.html" class="sidebar-link">Dynamic Programming</a></li><li><a href="/algorithm/applications-of-dynamic-programming.html" class="sidebar-link">Applications of Dynamic Programming</a></li><li><a href="/algorithm/kruskal-s-algorithm.html" class="sidebar-link">Kruskal's Algorithm</a></li><li><a href="/algorithm/greedy-algorithms.html" class="sidebar-link">Greedy Algorithms</a></li><li><a href="/algorithm/applications-of-greedy-technique.html" class="sidebar-link">Applications of Greedy technique</a></li><li><a href="/algorithm/prim-s-algorithm.html" class="sidebar-link">Prim's Algorithm</a></li><li><a href="/algorithm/bellman-ford-algorithm.html" class="sidebar-link">Bellman–Ford Algorithm</a></li><li><a href="/algorithm/line-algorithm.html" class="sidebar-link">Line Algorithm</a></li><li><a href="/algorithm/floyd-warshall-algorithm.html" class="sidebar-link">Floyd-Warshall Algorithm</a></li><li><a href="/algorithm/catalan-number-algorithm.html" class="sidebar-link">Catalan Number Algorithm</a></li><li><a href="/algorithm/multithreaded-algorithms.html" class="sidebar-link">Multithreaded Algorithms</a></li><li><a href="/algorithm/knuth-morris-pratt-kmp-algorithm.html" class="sidebar-link">Knuth Morris Pratt (KMP) Algorithm</a></li><li><a href="/algorithm/edit-distance-dynamic-algorithm.html" class="sidebar-link">Edit Distance Dynamic Algorithm</a></li><li><a href="/algorithm/online-algorithms.html" class="sidebar-link">Online algorithms</a></li><li><a href="/algorithm/integer-partition-algorithm.html" class="sidebar-link">Integer Partition Algorithm</a></li><li><a href="/algorithm/maximum-path-sum-algorithm.html" class="sidebar-link">Maximum Path Sum Algorithm</a></li><li><a href="/algorithm/maximum-subarray-algorithm.html" class="sidebar-link">Maximum Subarray Algorithm</a></li><li><a href="/algorithm/sliding-window-algorithm.html" class="sidebar-link">Sliding Window Algorithm</a></li><li><a href="/algorithm/sorting.html" class="sidebar-link">Sorting</a></li><li><a href="/algorithm/bubble-sort.html" class="sidebar-link">Bubble Sort</a></li><li><a href="/algorithm/merge-sort.html" class="sidebar-link">Merge Sort</a></li><li><a href="/algorithm/insertion-sort.html" class="sidebar-link">Insertion Sort</a></li><li><a href="/algorithm/bucket-sort.html" class="sidebar-link">Bucket Sort</a></li><li><a href="/algorithm/quicksort.html" class="sidebar-link">Quicksort</a></li><li><a href="/algorithm/counting-sort.html" class="sidebar-link">Counting Sort</a></li><li><a href="/algorithm/heap-sort.html" class="sidebar-link">Heap Sort</a></li><li><a href="/algorithm/cycle-sort.html" class="sidebar-link">Cycle Sort</a></li><li><a href="/algorithm/odd-even-sort.html" class="sidebar-link">Odd-Even Sort</a></li><li><a href="/algorithm/selection-sort.html" class="sidebar-link">Selection Sort</a></li><li><a href="/algorithm/pigeonhole-sort.html" class="sidebar-link">Pigeonhole Sort</a></li><li><a href="/algorithm/radix-sort.html" class="sidebar-link">Radix Sort</a></li><li><a href="/algorithm/shell-sort.html" class="sidebar-link">Shell Sort</a></li><li><a href="/algorithm/pancake-sort.html" class="sidebar-link">Pancake Sort</a></li><li><a href="/algorithm/searching.html" class="sidebar-link">Searching</a></li><li><a href="/algorithm/substring-search.html" class="sidebar-link">Substring Search</a></li><li><a href="/algorithm/breadth-first-search.html" class="sidebar-link">Breadth-First Search</a></li><li><a href="/algorithm/depth-first-search.html" class="sidebar-link">Depth First Search</a></li><li><a href="/algorithm/hash-functions.html" class="sidebar-link">Hash Functions</a></li><li><a href="/algorithm/travelling-salesman.html" class="sidebar-link">Travelling Salesman</a></li><li><a href="/algorithm/shortest-common-supersequence-problem.html" class="sidebar-link">Shortest Common Supersequence Problem</a></li><li><a href="/algorithm/knapsack-problem.html" class="sidebar-link">Knapsack Problem</a></li><li><a href="/algorithm/equation-solving.html" class="sidebar-link">Equation Solving</a></li><li><a href="/algorithm/longest-common-subsequence.html" class="sidebar-link">Longest Common Subsequence</a></li><li><a href="/algorithm/longest-increasing-subsequence.html" class="sidebar-link">Longest Increasing Subsequence</a></li><li><a href="/algorithm/check-two-strings-are-anagrams.html" class="sidebar-link">Check two strings are anagrams</a></li><li><a href="/algorithm/pascal-s-triangle.html" class="sidebar-link">Pascal's Triangle</a></li><li><a href="/algorithm/algo-print-a-m-n-matrix-in-square-wise.html" class="sidebar-link">Algo:- Print a m*n matrix in square wise</a></li><li><a href="/algorithm/matrix-exponentiation.html" class="sidebar-link">Matrix Exponentiation</a></li><li><a href="/algorithm/polynomial-time-bounded-algorithm-for-minimum-vertex-cover.html" class="sidebar-link">polynomial-time bounded algorithm for Minimum Vertex Cover</a></li><li><a href="/algorithm/dynamic-time-warping.html" class="sidebar-link">Dynamic Time Warping</a></li><li><a href="/algorithm/fast-fourier-transform.html" class="sidebar-link">Fast Fourier Transform</a></li><li><a href="/algorithm/pseudocode.html" class="sidebar-link">Pseudocode</a></li><li><a href="/algorithm/contributors.html" class="sidebar-link">The Contributors</a></li></ul></section></li></ul> </aside> <main class="page"> <div class="theme-default-content content__default"><h1 id="trees"><a href="#trees" class="header-anchor">#</a> Trees</h1> <h2 id="introduction"><a href="#introduction" class="header-anchor">#</a> Introduction</h2> <p><a href="https://en.wikipedia.org/wiki/Tree_(graph_theory)" target="_blank" rel="noopener noreferrer">Trees<span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a> are a sub-type of the more general node-edge graph data structure.</p> <p><a href="http://i.stack.imgur.com/BmT3t.png" target="_blank" rel="noopener noreferrer"><img src="http://i.stack.imgur.com/BmT3t.png" alt="A binary search tree."><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>To be a tree, a graph must satisfy two requirements:</p> <ul><li><strong>It is acyclic.</strong> It contains no cycles (or "loops").</li> <li><strong>It is connected.</strong> For any given node in the graph, every node is reachable. All nodes are reachable through one path in the graph.</li></ul> <p>The tree data structure is quite common within computer science. Trees are used to model many different algorithmic data structures, such as ordinary binary trees, red-black trees, B-trees, AB-trees, 23-trees, Heap, and tries.</p> <p>it is common to refer to a Tree as a <code>Rooted Tree</code> by:</p> <div class="language-cpp extra-class"><pre class="language-cpp"><code>choosing <span class="token number">1</span> cell to be called `Root`
painting the `Root` at the top
creating lower layer <span class="token keyword">for</span> each cell in the graph depending on their distance from the root <span class="token operator">-</span>the bigger the distance<span class="token punctuation">,</span> the lower the <span class="token function">cells</span> <span class="token punctuation">(</span>example above<span class="token punctuation">)</span>
</code></pre></div><p>common symbol for trees: <code>T</code></p> <h2 id="typical-anary-tree-representation"><a href="#typical-anary-tree-representation" class="header-anchor">#</a> Typical anary tree representation</h2> <p>Typically we represent an anary tree (one with potentially unlimited children per node) as a binary tree, (one with exactly two children per node). The "next" child is regarded as a sibling. Note that if a tree is binary, this representation creates extra nodes.</p> <p>We then iterate over the siblings and recurse down the children. As most trees are relatively shallow - lots of children but only a few levels of hierarchy, this gives rise to efficient code. Note human genealogies are an exception (lots of levels of ancestors, only a few children per level).</p> <p>If necessary back pointers can be kept to allow the tree to be ascended. These are more difficult to maintain.</p> <p>Note that it is typical to have one function to call on the root and a recursive function with extra parameters, in this case tree depth.</p> <div class="language-cpp extra-class"><pre class="language-cpp"><code> <span class="token keyword">struct</span> <span class="token class-name">node</span>
<span class="token punctuation">{</span>
<span class="token keyword">struct</span> <span class="token class-name">node</span> <span class="token operator">*</span>next<span class="token punctuation">;</span>
<span class="token keyword">struct</span> <span class="token class-name">node</span> <span class="token operator">*</span>child<span class="token punctuation">;</span>
std<span class="token double-colon punctuation">::</span>string data<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword">void</span> <span class="token function">printtree_r</span><span class="token punctuation">(</span><span class="token keyword">struct</span> <span class="token class-name">node</span> <span class="token operator">*</span>node<span class="token punctuation">,</span> <span class="token keyword">int</span> depth<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">int</span> i<span class="token punctuation">;</span>
<span class="token keyword">while</span><span class="token punctuation">(</span>node<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">if</span><span class="token punctuation">(</span>node<span class="token operator">-></span>child<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token keyword">for</span><span class="token punctuation">(</span>i<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>i<span class="token operator"><</span>depth<span class="token operator">*</span><span class="token number">3</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">" "</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"{\n"</span><span class="token punctuation">)</span><span class="token operator">:</span>
<span class="token function">printtree_r</span><span class="token punctuation">(</span>node<span class="token operator">-></span>child<span class="token punctuation">,</span> depth <span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">for</span><span class="token punctuation">(</span>i<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>i<span class="token operator"><</span>depth<span class="token operator">*</span><span class="token number">3</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">" "</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"{\n"</span><span class="token punctuation">)</span><span class="token operator">:</span>
<span class="token keyword">for</span><span class="token punctuation">(</span>i<span class="token operator">=</span><span class="token number">0</span><span class="token punctuation">;</span>i<span class="token operator"><</span>depth<span class="token operator">*</span><span class="token number">3</span><span class="token punctuation">;</span>i<span class="token operator">++</span><span class="token punctuation">)</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">" "</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token function">printf</span><span class="token punctuation">(</span><span class="token string">"%s\n"</span><span class="token punctuation">,</span> node<span class="token operator">-></span>data<span class="token punctuation">.</span><span class="token function">c_str</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
node <span class="token operator">=</span> node<span class="token operator">-></span>next<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">void</span> <span class="token function">printtree</span><span class="token punctuation">(</span>node <span class="token operator">*</span>root<span class="token punctuation">)</span>
<span class="token punctuation">{</span>
<span class="token function">printree_r</span><span class="token punctuation">(</span>root<span class="token punctuation">,</span> <span class="token number">0</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</code></pre></div><h2 id="to-check-if-two-binary-trees-are-same-or-not"><a href="#to-check-if-two-binary-trees-are-same-or-not" class="header-anchor">#</a> To check if two Binary trees are same or not</h2> <ol><li>For example if the inputs are:</li></ol> <p><strong>Example:1</strong></p> <p>a)</p> <p><a href="https://i.stack.imgur.com/Gzckc.png" target="_blank" rel="noopener noreferrer"><img src="https://i.stack.imgur.com/Gzckc.png" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>b)</p> <p><a href="https://i.stack.imgur.com/y2dy0.png" target="_blank" rel="noopener noreferrer"><img src="https://i.stack.imgur.com/y2dy0.png" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p><strong>Output should be true.</strong></p> <p><strong>Example:2</strong></p> <p>If the inputs are:</p> <p>a)</p> <p><a href="https://i.stack.imgur.com/C8jj7.png" target="_blank" rel="noopener noreferrer"><img src="https://i.stack.imgur.com/C8jj7.png" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p>b)</p> <p><a href="https://i.stack.imgur.com/BBfnO.png" target="_blank" rel="noopener noreferrer"><img src="https://i.stack.imgur.com/BBfnO.png" alt="enter image description here"><span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></a></p> <p><strong>Output should be false.</strong></p> <p>Pseudo code for the same:</p> <div class="language-cpp extra-class"><pre class="language-cpp"><code>boolean <span class="token function">sameTree</span><span class="token punctuation">(</span>node root1<span class="token punctuation">,</span> node root2<span class="token punctuation">)</span><span class="token punctuation">{</span>
<span class="token keyword">if</span><span class="token punctuation">(</span>root1 <span class="token operator">==</span> <span class="token constant">NULL</span> <span class="token operator">&&</span> root2 <span class="token operator">==</span> <span class="token constant">NULL</span><span class="token punctuation">)</span>
<span class="token keyword">return</span> <span class="token boolean">true</span><span class="token punctuation">;</span>
<span class="token keyword">if</span><span class="token punctuation">(</span>root1 <span class="token operator">==</span> <span class="token constant">NULL</span> <span class="token operator">||</span> root2 <span class="token operator">==</span> <span class="token constant">NULL</span><span class="token punctuation">)</span>
<span class="token keyword">return</span> <span class="token boolean">false</span><span class="token punctuation">;</span>
<span class="token keyword">if</span><span class="token punctuation">(</span>root1<span class="token operator">-></span>data <span class="token operator">==</span> root2<span class="token operator">-></span>data
<span class="token operator">&&</span> <span class="token function">sameTree</span><span class="token punctuation">(</span>root1<span class="token operator">-></span>left<span class="token punctuation">,</span>root2<span class="token operator">-></span>left<span class="token punctuation">)</span>
<span class="token operator">&&</span> <span class="token function">sameTree</span><span class="token punctuation">(</span>root1<span class="token operator">-></span>right<span class="token punctuation">,</span> root2<span class="token operator">-></span>right<span class="token punctuation">)</span><span class="token punctuation">)</span>
<span class="token keyword">return</span> <span class="token boolean">true</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
</code></pre></div><h4 id="remarks"><a href="#remarks" class="header-anchor">#</a> Remarks</h4> <p>Trees are a sub-category or sub-type of node-edge graphs. They are ubiquitous within computer science because of their prevalence as a model for many different algorithmic structures that are, in turn, applied in many different algorithms</p></div> <footer class="page-edit"><div class="edit-link"><a href="https://github.com/devtut/generate/edit/master/docs/algorithm/trees.md" target="_blank" rel="noopener noreferrer">Edit this page on GitHub</a> <span><svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" focusable="false" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg> <span class="sr-only">(opens new window)</span></span></div> <!----></footer> <div class="page-nav"><p class="inner"><span class="prev">
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