Questions tagged [binary-search-trees]

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How is the right, root, left order traversal called in a binary search tree?

In a Binary Search Tree you have the following orders for traversal: Left, Root, Right is called Inorder (or ascending order). Root, Left, Right is called Preorder. Left, Right, Root is called ...
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12 views

Time complexity for balancing an unbalanced binary tree

The question here is that: There is an unbalanced binary tree with n-nodes. What is the time complexity to balance the tree? The solution I thought of involved solving using Recursion where for the ...
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1answer
23 views

Is the tree shown a valid red-black tree?

I have made a red-black tree and I think that it is not valid. Could someone please verify? As far as I know, in red-black tree we also consider the leaf nodes at the NULLS of the visible leaf nodes ...
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17 views

RB trees from any balanced BST?

Given any perfectly balanced binary search tree, is it always possible to assign a coloring to the nodes so that it becomes a Red-Black tree? If so, how do you prove this, and if false, what would be ...
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AVL tree balance property states for the two subtrees of a node, their height can differ at most one. Why can't it be zero?

I was thinking that if they were equal, say they are required to be zero this would be enforce the balance property more effectively. Can anyone explain why 1 is a satisfactory rather than just them ...
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3answers
118 views

Why is Binary Heap never unbalanced?

My professor asks this question: Binary Search tree has Rotation Method to prevent it from degenerating into a linear structure (unbalanced tree). Why is there no need for such method for Binary Heaps?...
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1answer
52 views

Which binary search trees have constant time rebalancing time at min/max?

Given that I'm already at either the min or the max node of a binary search tree, which balanced variant would require only constant time bottom-up rebalancing after an update (add new min/max, or ...
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25 views

Finding Binary Search Tree Height, What happens to duplicates?

I'm going over past exam papers with this question: The answer to a). will be (height of 5): For b.): I assume the height function will be to find the height of the tree. So, I will have to compare ...
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38 views

Tracing BST Pseudocode for Insertion

I'm having real trouble tracing some pseudocode for insetion into a Binary Search Tree. I need to understand this in order to be able to teach it, but I think there are flaws in the material. For ...
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2answers
39 views

Algorithm for searching in BST with only <

How could one construct an algorithm for finding a node in a binary search tree that only requires the presence of $<$ on the key type. The ones I can easily also requires $=$.
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1answer
30 views

Augmenting a tree such that we preserve the insertion operation optimal runtime

Suppose we are given a red-black tree with $n$ vertices with distinct keys and we want to store, as addition information in each vertex $v$, the biggest key out of the keys that are smaller than $v$ (...
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1answer
58 views

Proof that a subtree of a red-black tree has no more than $\frac{3n}{4}$ nodes

I have a red-black tree with $n$ nodes, rooted at $x$. How can I prove or disprove that the number of nodes in any subtree of $x$ (including the root of the subtree) will never be greater than $\frac{...
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avl trees rotations question

As you can see in this tree its unbalanced at the root with a balanced factor of $-2$. You can also perform a Right left rotation or a Right Right rotation. Which do you have to pick in this case?
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1answer
61 views

fixing a slightly broken BST tree

I'm stuck on this question, a BST is slightly broken only if there is a node with value x in which : there is at least one value less than x on its right branch or atleast one value greater than x on ...
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1answer
272 views

Binary Search Tree – Number of comparisons when element is not found

I am confused regarding Binary Search Trees (BST) when an element does not exist in the tree. For example, to search for element "6", would it take 5 comparisons to search for this element? My ...
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1answer
309 views

AVL Tree rotations : How to balance below AVL Tree with imbalance at root node

How to balance this AVL tree after inserting node 5, using left/right rotations as indicated in the AVL tree tutorial. I have tried applying both double rotations but with no luck. Either of the ...
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1answer
56 views

Searching in a Binary Search Tree

I'm studying Binary Search Trees (BST) and I would like to verify that my understanding of BSTs is correct. For example, let S = [17, -10, 7, 19, 21, 23, -13, 31, 59]. Binary Search Tree for S, with ...
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BST - Sufficient condition for connecting a node to a parent

Let's assume that we have a binary search tree with node Y that hasn't a right child and for whom a successor exists in the tree. I want to prove that if I insert a node X into the tree and node X is ...
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1answer
90 views

How to find the number of Binary Search Trees with given number of nodes and leaves?

With 7 nodes of distinct values (unique), how many Binary search trees (BST) can be formed such that: Exactly $1$ leaf node(s) present? Exactly $2$ leaf nodes present? I was able to solve the first ...
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117 views

Deriving the average depth for a randomly generated binary search tree

If $D(n)$ is the internal path length (sum of the depths of all nodes) for some tree $T$ with $n$ nodes then we have the following recurrence relation: $$D(n)=D(i)+D(n-i-1)+N-1$$ where I simply taken ...
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2answers
91 views

Calculation of Inorder Traversal Complexity

I want to analyze complexity of traversing a BST. I directly thought that its complexity as $O(2^n)$ because there are two recursive cases. I mean $T(n) = constants + 2T(n-1)$. However, AFAI research ...
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1answer
90 views

All the possible inputs for a given AVL tree

Given an AVL tree,what are the possible inputs so that the same given tree is formed(please dont mention brute force technique)?
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1answer
57 views

The validity of the potential function for splay tree

The paper "Self-Adjusting Binary Search Trees" defines (Page 658) the potential function for analyzing the amortized cost of a sequence of $m$ splay operations as the sum of the ranks of all nodes in ...
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1answer
202 views

BST representation of Hash Tables

I'm reading this book Cracking Coding Interview. In this book author is speaking about BST representation of Hash Tables. I googled a lot for BST Representation of Hash Table. Code in C++ but didn't ...
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1answer
712 views

Average height of a BST with n Nodes

I have to find the maximum, minimum, and average height of a BST with n nodes. After doing some researching I found that the maximum height is $n-1$ and the minimum height is $\log_2(n+1)-1$. My ...
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0answers
55 views

Upper bound on the average path length in binary search tree

I have been reading the chapter 6.2.2 in Knuth's book about lower and upper bound on the average path length in binary search tree. And I have problems with understanding small details of Theorem M (...
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45 views

Example of binary trees with maximum rotation distance

In the 1988 paper Rotation Distance, Triangulations, and Hyperbolic Geometry, Sleator, Tarjan and Thurston show that for any pair of $n$-node binary trees, the maximum rotation distance between them ...
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1answer
38 views

What is the average size of a jump in a binary tree?

Assume that a full binary tree is layed out in memory recursively in the following way. First the Root followed by Tree representation of left subtree followed by tree representation of right subtree. ...
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2answers
929 views

If inorder traversal of a tree is in ascending order will the tree definitely be a BST?

For a binary search tree (BST) the inorder traversal is always in ascending order. Is the converse also true?
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Splay Tree - Insert Permutation

Let $T$ be a Splay Tree. For a given permutation $\sigma$ on a set $S = \{1,2,3,...n \}$ we defined the following function: ...
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1answer
79 views

Binary Tree Node Insertion

I was trying to implement a Binary Search Tree using this article as a reference: Binary Search Tree in JavaScript. I was thinking especially about the node insertion method. Here's my ...
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1answer
186 views

All possible Red Black Trees with this set {1,2,3,4,5}

I have to write all possible Red Black Trees which can represent these 5 numbers {1,2,3,4,5}. Now we have 120 ways to write 1,2,3,4,5 ...
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1answer
119 views

how does rotation works in AVL trees and what is a good way to understand it?

If we consider this tree with T1 and T2 as subtrees, and we want to rotate on x (rotating the edge between T1 and x), what is the result? how does it work then? Does the x stay in its place and T1 ...
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Why use binary search trees when hash tables exist?

Hash tables perform lookup, insertion, and deletion in O(1) time (expected and amortized) while the different variants of binary search tree (BST) - treap, splay, AVL, red-black - offer at best O(log ...
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101 views

Time complexity proof of finding the $i$ object in binary search tree is $O(h+i)$ by inorder

I'm looking for time complexity proof of finding the $i$ object in binary search tree is $O(h+i)$ by inorder run. when $h$ is the height of the tree.
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1answer
798 views

Number of binary search trees with maximum possible height for n nodes

I'm using the definition of the height of a tree as the longest possible path from the root to a leaf by its number of edges, e.g. a tree of 2 nodes has a height of 1. With that in mind, what would ...
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1answer
56 views

Complexity for find pairs with sum - BST

I have written an algorithm for find all the pairs in a BST which have a given sum. ...
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0answers
26 views

Looped binary sorting tree

I want to add a points one by one into sorting tree. I have less then function to order them by angle from absciss: for points $\mathbf p_i = (x_i,y_i)$ and $\mathbf p_j = (x_j,y_j)$ it just sign of ...
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1answer
245 views

Simulating AVL Tree Right, then Left rotation

I have the following AVL tree and want to AVL-INSERT a node 5 into the tree. Since the middle branch will be unbalanced, I'm guessing that it will require a right rotation, then a left rotation to ...
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1answer
39 views

k-means clustering with efficient point lookup?

What's an algorithm for $k$-means clustering, in particular an online algorithm (you can stream new points to it), such that once the size of the set of clusters $k$ becomes large, we can still ...
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1answer
454 views

Time complexity of creating the unique binary tree from given inorder and preorder (or postorder) traversal sequences

Given inorder and preorder (or postorder) traversal sequences of a binary tree balanced binary tree binary search tree of n nodes, what is the time complexity of creating the respective unique tree.
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How many similar binary search trees are made from different permutations?

If I have numbers ranging from 1 to n, and I generate all the permutations of these numbers. Now, I create Binary search tree from these permutations. For a value n i want to know how many BSTs would ...
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1answer
29 views

Generate a unique number for a set of sequences of letters

A sequence of English letters are given, every sequence forms a word and we know that the size of each word is at most $15$ and the number of words is at most $50000$. For a given input word $w$ I ...
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32 views

AVL Tree Inner Nodes

If k is the height of an AVL Tree, the minimal number of inner nodes is: N(k) = N(k-1) + N(k-2) +1. Why is this formula true?
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Which of the following sequences could not be the sequences of nodes examined in a binary search tree?

Suppose that we have numbers between $1$ and $1000$ in a binary search tree and want to search for the number $363$. Which of the following sequences could not be the sequences of nodes examined <...
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2k views

Time complexity of searching an element in Binary Search Tree [closed]

I want to know what will be the time complexity to search, insert and delete an element in a) balanced binary search tree. b) unbalanced binary search tree.
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1answer
331 views

Rooted Binary Tree - Chain Nodes Algorithm

The question is that I have to give a recursive algorithm in pseudocode, that counts the number of chain node in a rooted binary tree. Chain node is defined as a node that has only one child and that ...
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113 views

An AVL tree is a BST, but AVL tree can be not complete, then why do we define AVL tree a BST?

An AVL tree is a BST, and a BST is a complete tree, but AVL tree doesn't necessarily have to be complete. Thus, why do we define AVL tree a BST?
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1answer
186 views

Is there an example of an AVL tree that is taller than a binary heap?

I am curious because I took a quiz and answered false to the question: "The height of an AVL tree may be greater than the height of a binary heap with the same elements." I have been trying to think ...
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1answer
449 views

When to rebalance the AVL tree?

Consider we have an AVL tree. Its left subtree A has disbalance coefficient has value +1 (meaning the right subtree has greater depth). Its right son has disbalance coefficient -1 (meaning left ...