Questions tagged [binary-search-trees]

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2 answers
137 views

Recursive formula for height of BST

Let $H(n)$ be the average height of a BST with nodes from ${1,...,n}$. I think that $$H(n) = \frac{1}{n}\sum_{i = 0}^{n-1}\left[\text{max}(H(i), H(n-1 -i)) + 1\right]$$ But I don't know how to prove ...
1 vote
2 answers
93 views

What factors of the integer dataset being sorted can I change, in order to compare two sorting algorithms?

I am comparing two comparison and binary data structure based sorting algorithms, the Tree Sort, and the Heap Sort. I am measuring the time taken for both algorithms to sort an increasing size of an ...
0 votes
1 answer
53 views

Time Complexity: Determining if a binary tree is balanced

I found an algorithm for determining if a binary tree is height-balanced. It Gets the height of left and right subtrees using dfs traversal. Return true if the difference between heights is not more ...
0 votes
2 answers
744 views

Binary Search Tree Updating

How can I update values in Binary Search Tree without affecting its properties (all the nodes in the left subtree have values that are less than the value of the root node and all the nodes of the ...
1 vote
1 answer
58 views

Does a sorted sequence from in-order traversal imply a binary tree is a BST?

An in-order traversal of a binary search tree (BST) produces a sorted sequence. I wonder, if we perform an in-order traversal of a binary tree and obtain a sorted sequence, does that imply that the ...
0 votes
0 answers
23 views

Create a class or structure like union of ranges

How to create a structure which acts as union of ranges. In that structure new ranges can be inserted beforehand and then some queries are asked to find out if the given point is covered by any of the ...
0 votes
2 answers
828 views

Leaf nodes of B+ Tree

I have a b+ tree and i want to find the record associated with a specific key Ki. So i run the b+ tree search algorithm. If a certain node in the search path is a leaf and K=Ki, then the record exists ...
0 votes
1 answer
102 views

Find two nodes in a BST such that the root's key is the average of their keys without extra space in $\theta(n)$ worst case time

We can do this in $\theta(n^2)$ time if we calculate the average of all couples of nodes in the tree and compare it to the root, but this is too much time. We can do this in linear time but with extra ...
0 votes
1 answer
37 views

Is binary tree balanced if and only if the morris traversal of the tree produces ordered list?

I'm trying to check if the binary tree is binary search tree. My idea is to use Morris traversal. Intuitively a binary tree is balanced iff Morris traversal produces a sorted threaded linked list. The ...
0 votes
1 answer
33 views

Splay Trees - Sequential Access Theorem & lower bound for comparison-based sorting

The following theorem was proven by R.E. Tarjan in 1984: Theorem (Sequential Access Theorem). If we access each of the nodes of an arbitrary initial tree once, in symmetric order, the total time ...
2 votes
1 answer
131 views

Visualizing How of KD-tree Data Structure Splits Space

I am trying to understand how KD-tree works when we insert a node and how it splits the xy plane, please. Below $[5, 4]$ splits the xy-plane into left and right parts while $[2,6]$ splits it into top ...
0 votes
0 answers
38 views

Can a Red Black Tree have a black node with a black descendant? Is this image wrong?

How the story begins? I was learning about Red Black Trees on my own. When one opens the Wikipedia article, it can be read: (Conclusion) If a node N has exactly one child, it must be a red child, ...
5 votes
0 answers
76 views

Completeness of red-black tree operations

Red-black trees are defined to have the following invariants: The nodes are in sorted order (it is a binary search tree). The root is black, and leaves are black. Every red node has black children. ...
1 vote
1 answer
55 views

Merge K BST of N elements in total into a single RBT in O(N log K) time

I have the following question to solve; Given $K$ BST consisting of $N$ total elements, show how you can create a Red Black Tree in $O(N\log K)$ time. I had the following idea but it falls on the ...
1 vote
2 answers
91 views

Merge two binary search trees

Consider two binary search trees T1 and T2, each with height h, with all values in T1 less than all values in T2. I want to merge these both trees to get a new binary search tree of height at most h+1 ...
0 votes
0 answers
71 views

Deletion of a node from a BST

Placed exactly the code and the explanation of the book : Introduction to Algorithms Third Edition In order to move subtrees around within the binary search tree, we define a subroutine TRANSPLANT, ...
0 votes
0 answers
25 views

How many times is a node rotated towards the root in a weight-balanced tree?

In this paper they prove that the number of rotations after doing $n$ insertions or deletions to a weight-balanced tree is $O(n)$ (when starting from an empty tree). What isn't clear to me though is ...
1 vote
1 answer
247 views

Interval Tree by Augmenting an AVL Tree

According to Wikipedia: An augmented tree can be built from a simple ordered tree, for example a binary search tree or self-balancing binary search tree, ordered by the 'low' values of the intervals. ...
1 vote
1 answer
141 views

Two statements about optimal binary search trees

This is a paragraph from the book CLRS: What we need is known as an optimal binary search tree. Formally, we are given a sequence $K = (k_1, k_2, ..., k_n)$ of $n$ distinct keys in sorted order (so ...
1 vote
1 answer
77 views

What is the minimal number of nodes with the right subtree in a height-balanced BST?

I have a binary search tree of size $N$. The tree is height balanced: difference of heights of a node's subtrees is no more than 1 (true for RB or AVL trees). $K$ is the number of nodes that have a ...
1 vote
1 answer
113 views

Randomly generated binary search trees case comparison

Although not an assignment, just out of curiosity; I am trying to compare a two cases A scenario where I pick a tree out of the set of possible binary search trees on the keys $1,2,\ldots,n$, with ...
1 vote
1 answer
1k views

Prove correctness of in-order tree traversal subroutine

I'm trying to prove that in-order tree traversal prints the keys in sorted order. It's shown here, but what I want is to prove correctness using ordinary induction. Claim: For any n-node subtree, ...
2 votes
0 answers
230 views

Is there a way to parallelise find and inserts for a binary search tree?

Background: I'm working on a data structure benchmark tool to benchmark insert and search time and I am trying to improve my own implementation of a BST to support parallelism. I have implemented a ...
0 votes
1 answer
135 views

Did I invent a new data structure?

I needed to implement a priority queue for a project I'm working on and had this idea. In a priority queue BST implementation wouldn't it be more efficient if the poll node pointed to its parent since ...
0 votes
1 answer
137 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 ...
0 votes
0 answers
157 views

How does depths of nodes change after left-rotation in a BST (Exercise question from Cormen)

Let a,b,c be arbitrary nodes in the subtrees $\alpha$, $\beta$, $\gamma$, respectively, in the left tree of Figure 13.2 (that is given below). How do the depths of a,b,c change when a left rotation is ...
7 votes
3 answers
4k 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?
0 votes
0 answers
31 views

What is the gold-standard description of the 2-3 tree (search, insert, delete)?

After several hours of frustration, I finally realized that the definition of two-three trees aren't standard (my lecture notes, this video, this other video, and Wikipedia) are different. My lecture ...
2 votes
1 answer
119 views

Solution Verification: How does the postorder traversal of a BST change after rotating left?

Given a BST $T$, $x$ is a random node in it and $y$ is the right child of $x$. How does the PostOrder traversal of BST $T$ change after we rotate the tree left on node $x$? In which cases does the ...
0 votes
0 answers
70 views

Complexity of finding the kth smallest element of all the elements in two order statistics binary search trees

What is the time complexity of finding the kth smallest element of all the elements in two order statistics binary search trees? An order statistics tree is a binary search tree where the size of a ...
0 votes
1 answer
544 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? ...
1 vote
1 answer
329 views

Splay Tree: Repeatedly searching for the same key that's not in the Tree

In a splay tree, doing $m$ sequential search operations for the same key that is in the tree has a time complexity in $O(n+m)$ where n is the number of nodes in the tree. Since the first search has a ...
0 votes
1 answer
87 views

I think I have discovered a new sorting algorithm using binary search tree [closed]

If we some how transform a Binary Search Tree into a form where no node other than root may have both right and left child and the nodes the right sub-tree of the root may only have right child, and ...
2 votes
2 answers
885 views

What is the time complexity of adding to a BST if we are to maintain balance

If we have a BST but want to keep it balanced, how much more expensive does adding an element to it become? Clearly adding an element (without maintaining balance) is of time complexity O(log(n)), as ...
0 votes
0 answers
56 views

Red Black Tree: number of internal nodes vs leaf nodes

Given a generic Red Black Tree with n nodes is correct to say that the number of internal nodes is ⌊n/2⌋ and the number of leaf nodes is ⌊n/2⌋ + 1 ?
0 votes
1 answer
142 views

Depth-first search (DFS) time complexity for a Red-Black Tree

If we indicate n as the number of nodes of a Red-Black Tree, which is the time complexity of a DFS algorithm that analyzes only the internal nodes of the Tree? I think that the complexity is O(n), but ...
3 votes
1 answer
147 views

Using pre-,post-, and in-order indexes to find information about a Binary Search Tree

Recently I have been studying ways of traversing a BST (in python), and have collided with the terms pre-order, post-order and in-order. I believe that I understood the three terms pretty well, and ...
3 votes
1 answer
279 views

finding an algorithm for creating a priority search tree in linear time with presorting

A priority search tree is a binary tree satisfying the following: every node $u$ stores a point $p_u = (x_u,y_u)$ every nonleaf $u$ stores an x-coordinate $x_u'$ called the split-line coordinate. If $...
0 votes
1 answer
96 views

BST subtree value range

Suppose we have a node x in BST, and let max and min be the largest and smallest keys in the subtree rooted at x respectively. Prove that for any node n outside this subtree, the key of n is either ...
4 votes
1 answer
670 views

Big O vs. Big Theta for AVL tree operations

On the Wikipedia page for AVL trees, the time/space complexity for common operations is stated both for average case (in Big Theta) and worst case (in Big O) scenarios. I understand both Big O and Big ...
0 votes
1 answer
47 views

n search operations on an arbitrary Splay tree

For an arbitrary spay tree with n nodes, if we perform n find operations, is there a way of generalizing what the tree would ...
1 vote
1 answer
87 views

Least-balanced possible red-black tree of n distinct nodes

Let's say we have a red-black tree of $n$ total nodes where all keys are distinct. The subtree rooted at the root node's left child has $n_L$ nodes, and similarly the subtree rooted at the root node'...
0 votes
1 answer
30 views

Which particular data structure should I use if I want a persistent balanced search tree?

As title, I'm trying to implement a text editor with the rope data structure, which is backed by binary search tree. Since I want it to have persistent undos, the underlaying data structure should ...
0 votes
1 answer
92 views

What's the sum of heights of a random binary search tree

What's the sum of heights of a random binary search tree? By a random binary search tree, I mean the usual definition: you have $n$ keys to be inserted, and all Permutations are equally likely. The ...
0 votes
1 answer
477 views

Height of AVL Tree

I found an AVL tree implementation on the internet and experimented: For a tree with node count of 2^20, the minimal and maximal tree heights are 16 and 24. While these heights are lg(n)-ish, I am ...
2 votes
4 answers
1k views

Why is the number of comparisons in a BST missing key lookup about 2 ln N?

In (An Introduction to the Analysis of Algorithms) by Philippe Flajolet and Robert Sedgewick it's written that: Insertions and search misses in a BST built from N random keys require ~ 2 ln N (about 1....
0 votes
0 answers
26 views

Two variants of the BST remove algorithm

I have seen two variants of the BST remove algorithm in the case where the node to be removed has both children. Variant 1: Replace the node with the right-most child in the left subtree Variant 2: ...
1 vote
3 answers
440 views

How does the Inorder-tree-walk algorithm move to a different node after hitting a leaf?

A friend and I did the Inorder-tree-walk with pen and paper. We both can't figure out how the algorithm would move 'up' the tree again upon hitting a leaf: We are using the algorithm as described by ...
0 votes
1 answer
179 views

How many different (full/complete) in-order binary-trees do exist?

Given be a binary tree whose elements printed in-order results in [1,2,3,4]. Q1: How many different binary-trees do exist? Q2: How many different complete binary-...
2 votes
1 answer
126 views

How to join two Scapegoat Trees in O(log n) time?

I am working on some binary-search-tree research and was surprised to find no mention of an algorithm to join two Scapegoat Trees. This is where two trees $L$ and $R$ are joined to create a single ...