Fannie  Zemlak

Fannie Zemlak

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Binary Trees

This week while diving further into coding interview questions, I took a deeper look at data structures, particularly trees. In programming, trees are a non linear data structure, they instead maintain data in a hierarchical way. Similar to other structures however, they are comprised of a bunch of nodes that contain pieces of data. Every tree has a “root node” and from this root node, children nodes are branched off and connected by edges. Once we traverse to the bottom of a tree, these bottom level nodes are called leaves. You can picture this data structure as a real life tree but the properties are upside down, with the root being on the top vs the bottom. Every node, other than this root has one parent, however they can have a varying amount of children nodes. Children can then have sibling nodes which are horizontal to them, as well as grandparent nodes which are two levels up.

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Tree Terminology

  • **Root — **The top node in a tree.
  • **Child — **A node directly connected to another node when moving away from the root.
  • **Parent — **The converse notion of a child.
  • **Siblings — **A group of nodes with the same parent.
  • **Leaf — **A node with no children.
  • **Edge — **The connection between one node and another.

Following the names of parent, sibling and child, a common real life example of this data structure is a family tree. Even if you aren’t familiar with trees, as a programmer you’ve likely worked with one before, the DOM. The DOM also organizes its data (the elements of a page) in a hierarchical way. There are, however, many different types of trees in programming and in this article I will quickly summarize some of the most common Binary trees. The different types of trees we’ll discuss include Binary SearchAVL, and **Red-Black **trees. Each tree type has its own use case and distinctions.

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Starting with the basic Binary tree, its main distinction is that each node can have, at most, 2 children nodes. This tree is one of the most common ways you’ll see data organized in a tree structure and all the other trees we’ll cover are subsets of this tree. A binary tree can be considered either complete, full or perfect. A perfect binary tree is when all nodes have 2 children nodes and all leaves end at the same level. An example of a complete binary tree has every level other than maybe the last completely filled. In a full binary tree every node has either 0 or 2 children. A binary tree is the most basic form of a tree data structure. and there are many more types than just the 3 we’ll cover.

A binary search tree (BST) is an extension of a binary tree with additional restrictions. Each node still has a max of two children like the binary tree, however the values and where they’re placed matter. The left child of the parent should always be less than or equal to that parent node. And vice verse for the right side, it should always be greater than or equal to the parent. The fact that the tree keeps its data organized makes searching operations entirely more efficient, hence the name binary search.

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An AVL tree is a self balancing binary search tree. The name comes from the creators Adelson-Velshi and Landis who helped create the first tree that balances dynamically. A balancing factor is added to each node in the tree based on if the tree is balanced or not (-1, 0, 1). In this tree, the heights of the two children subtrees can’t differ by more than 1. If the tree gains a new node and it causes the heights to differ by more, then it’ll be rotated to make sure the tree remains balanced. Operations such as access, removal and insertion with an AVL tree all maintain a good Big O runtime of O(log n). The tree is most beneficial when used for lookup operations.

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Finally red-black trees are very similar to AVL trees. A red-black tree is also self balancing as well as a subset of the binary search tree. Each node in this tree is either red or black which is useful to ensure the tree remains somewhat balanced. When new nodes are inserted or deleted, the nodes will be rearranged to maintain its balance. The balancing isn’t perfect but allows us to still expect a decent runtime. This tree maintains a solid time complexity of O(log n).

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In conclusion, trees are known as one of the most powerful data structures. The primary advantages include how it easily displays data through organization, the efficiency it provides in operations such as searching and insertion, and its flexibility. Trees are most useful to show relation among various nodes, especially in these binary trees where there’s a limit of two children. The categories of different trees include Binary trees, B-trees, Heaps, General trees, Multiway trees, Space-partitioning trees and Application-specific trees. There are numerous types in each tree category and to memorize them all would take an incredible amount of time. However, I feel taking the time to understand some common ones, definitely enables you to get a strong feel for use cases and possible coding implementations. Trees aren’t native to JavaScript, but definitely something you should consider in future coding projects!

#data-structures #coding #medium

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Binary Trees
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Check if all the Nodes in a Binary Tree having common values are at least D distance apart

Given a Binary Tree and an integer D, the task is to check if the distance between all pairs of same node values in the Tree is ? D or not. If found to be true, then print Yes. Otherwise, print No.

Examples:

Input:_ D = 7 _

                1
              /   \ 
             2     3
            / \   /  \ 
           4   3  4   4

Output:_ Yes _

Explanation:

_The repeated value of nodes are 3 and 4. _

_The distance between the two nodes valued 3, is 3. _

_The maximum distance between any pair of nodes valued 4 is 4. _

Therefore, none of the distances exceed 7

Input:_ D = 1 _

          3
         / \
        3   3
             \
              3

Output:_ No _

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

**Approach: **

The idea is to observe that the problem is similar to finding the distance between two nodes of a tree. But there can be multiple pairs of nodes for which we have to find the distance. Follow the steps below:

  1. Perform the Post Order Traversal of the given tree and find the distance between the repeated pairs of nodes.
  2. Find the nodes that are repeated in the tree using unordered_map.
  3. For each repeated node of a particular value, find the maximum possible distance between any pair.
  4. If that distance is > D, print “No”.
  5. If no such node value is found having a pair containing that value, exceeding **D, **then print “Yes”.

#greedy #recursion #searching #tree #binary tree #frequency-counting #postorder traversal #tree-traversal

Double Order Traversal of a Binary Tree

Given a Binary Tree consisting of** N** nodes, the task is to print its Double Order Traversal.

Double Order Traversal_ is a tree traversal technique in which every node is traversed twice in the following order: _

  • Visit the Node.
  • Traverse the Left Subtree.
  • Visit the Node.
  • Traverse the Right Subtree.

Examples:

Input:
        1
      /   \
     7     3
    / \   /
   4   5 6
Output: 1 7 4 4 7 5 5 1 3 6 6 3 

Input:
        1
      /   \
     7     3
    / \     \
   4   5     6
Output: 1 7 4 4 7 5 5 1 3 3 6 6

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

The idea is to perform Inorder Traversal recursively on the given Binary Tree and print the node value on **visiting a vertex **and after the recursive call to the left subtree during the traversal.

Follow the steps below to solve the problem:

#data structures #recursion #tree #binary tree #inorder traversal #data analysis

Construct a Maximum Binary Tree from two given Binary Trees

Given two Binary Trees, the task is to create a Maximum Binary Tree from the two given binary trees and print the Inorder Traversal of that tree.

What is the maximum Binary Tree?

_The __maximum binary __is constructed in the following manner: _

_In the case of both the Binary Trees having two corresponding nodes, the maximum of the two values is considered as the node value of the Maximum Binary Tree. _

_If any of the two nodes is NULL and if the other node is not null, insert that value on that node of the Maximum Binary Tree. _

Example:

Input:
Tree 1                Tree 2
   3                    5 
  / \                  / \
 2   6                1   8 
/                      \   \ 
20                      2   8 
Output: 20 2 2 5 8 8
Explanation:
          5
         / \
        2   8
       / \   \
      20   2   8

To construct the required Binary Tree,
Root Node value: Max(3, 5) = 5
Root->left value: Max(2, 1) = 2
Root->right value: Max(6, 8) = 8
Root->left->left value: 20
Root->left->right value: 2
Root->right->right value: 8

Input:
       Tree 1            Tree 2 
         9                 5
        / \               / \
       2   6             1   8
      / \                 \   \
     20  3                 2   8
Output:  20 2 3 9 8 8
Explanation:
          9
         / \
        2   8
       / \   \
      20  3   8

#data structures #mathematical #recursion #tree #preorder traversal #tree traversals

Remove all leaf nodes from a Generic Tree or N-ary Tree

Given a Generic tree, the task is to delete the leaf nodes from the tree.

** Examples:**

Input: 
              5
          /  /  \  \
         1   2   3   8
        /   / \   \
       15  4   5   6 

Output:  
5 : 1 2 3
1 :
2 :
3 :

Explanation: 
Deleted leafs are:
8, 15, 4, 5, 6

Input:      
              8
         /    |    \
       9      7       2
     / | \    |    / / | \ \
    4  5 6    10  11 1 2  2 3
Output:  
8: 9 7 2
9:
7:
2:

**Approach: **Follow the steps given below to solve the problem

  • Take tree into the vector.
  • Traverse the tree and check the condition:
  • If current node is leaf then
  • Delete the leaf from vector
  • Else
  • Recursively call for every child.

Below is the implementation of the above approach:

#data structures #recursion #tree #n-ary-tree #tree-traversal #data analysis