# Questions tagged [binary-search-trees]

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### 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. ...
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### Understanding the Transition points of a BST

I'm trying to understand the definition of Transition point of a BST, as given in Demaine, Erik D., et al. "Dynamic optimality-almost." SIAM Journal on Computing 37.1 (2007): 240-251 ...
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### If I randomly swap pairs of inputs how unbalanced will a binary search tree become?

Treesort sorts input by putting input into a binary search tree and then flattening the tree. In some cases the tree can become unbalanced and so a self-balancing tree is required. There is no ...
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### 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 ...
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### Algorithm to delete BST nodes with duplicated values

In a binary search tree the following must hold: Greater keys are in the right-subtree Smaller or EQUAL keys belong to the left-subtree All the algorithms I found to delete a node start by finding ...
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### Reasoning for awkward style of TreeMap.getHigherEntry

Consider a binary search tree (AVL, red-black, whatever). The goal "find the least key strictly greater than the specified one" ("upper_bound" in C++ STL, higherEntry//higherKey in Java, etc.) can be ...
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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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### What would be the probabilty of a randomly generated tree to be a Red-Black Tree

The question is not related to the homework I was working on a homework, and the specification was to generate a random tree with n elements(n being in the thousands for the assignment) and asked me ...
1 vote
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### Finding height of binary search tree

Suppose given a binary search tree $T$ with $n$ nodes with depth $h$. We did $n$ times search with cost $c_i$ for search $i^{th}$ search on $T$ ,and $\sum_{i=1}^{n}c_i=O(n\log n)$. what we can say ...
1 vote
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### Splay tree amortized analysis cost using Access Lemma

Currently studying for an algorithms exam and I came across this question and solution, but I can't understand the solution where it references nodes of depth less than $4\log n$ and not restructuring....
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1 vote
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### 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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1 vote
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### 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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1 vote
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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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### 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 ...
1 vote
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### Read nodes of a BST in blocks of size $k$ and traverse it in $\mathcal{O}(log_kn)$

This describes how one can neatly store a binary search tree as an array. I'm looking for a way to store a BST that will allow me to traverse any root to leaf path by loading $\mathcal{O}(log_kn)$ ...
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1 vote
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### Day-Stout-Warren algorithm for balancing BST. How does vine to tree work?

I was trying to understand the dsw algorithm for balancing a binary search tree in-place using this virgina tech page. The wikipedia page and vt page are more or less similar. I am having a hard time ...
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### 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 ...
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, ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ?
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### 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: ...
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### Rebalance the following AVL tree after inserting G. You need to show the middle step if it happens. Briefly explain the operations

I recently learned AVL Trees but I still lack a complete understanding of the concept. I understand that an AVL tree is a Binary search tree that checks for the height of the tree, but I still don't ...
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### 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 ...
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### Calculating the expected height of a randomly built binary search tree

I need to calculate the expected height of a randomly built binary search tree, BST, with 4 different keys: $x < y < z < w$ According to Catalan numbers, there are 14 possible trees, 8 with ...
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### BST deletion of node with 2 children by copying right child to predecessor

I try to delete a node, $z$, with 2 children in binary search tree by exchanging $z$'s right child and $z$'s predecessor, $y$, right child, which is NULL because $y$ has no right child, if it had then ...
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### Suggest a Data Structure that support the following operations with time complexity O(log(n))

I’m looking for a data structure that supports that store the salaries of it’s employees. Insert(e) – Insert employee e into the data structure. AvgDecile(k) – Returns the average salary of the k’th ...
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### Probability that BST has exact height

Consider keys $[ 1 \ldots n]$. We want to calculate probability that BST tree has height = $h$. (We assume that distribution is uniform over all $C_{n}$ trees, where $C_n$ - n-th Catalan number). ...
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### Parent Node in BST

Is it a good idea to store a pointer to the parent node in a binary search rather than to find the parent node several times? ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### How to find point of maximum overlap in both X and Y Axis?

I was given a array R of n rectangles [each element in the array is a bottom left point(x1,y1) and a top right point (x2,y2)] , parallel to the axis, and an array P of n points [each element is a ...
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### IO efficiency of a complete binary search tree

I was given this as a homework exercise, yet I don't know whether my solution is sound: Given a complete binary search tree where blocks are formed in order of the value of the nodes. What would ...
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### Find keys between $x$ and $y$ in binary search tree

Given $x$ and $y$, I want to find keys $k$ such that $x<k<y$, in a binary search tree. Can this be done in time $O(n + h)$, where $n$ is the number of keys between $x$ and $y$, and $h$ is the ...
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### 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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### post order for binary search tree

I got this as the post order sequence but the answer says it is wrong. I do get a bit confused with the post order logic as well. 8 11 10 9 13 16 18 15
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### 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)?
### Binary search trees: multiple Successor() calls
Show that, given a tree node a, the time complexity of calling k times to Successor() is $O(k+h)$, where $h$ is the tree height. ...