# Questions tagged [balanced-search-trees]

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### 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 ...
27 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. ...
64 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 ...
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### Are AVL&RB Trees without additional storage for balance information in each node feasible?

One advantage claimed for scapegoat trees over other balanced trees like AVL or red-black(RB trees - just mentioning AVL henceforth) is not needing to store additional balance information. But can't ...
166 views

### Lookup complexity in B-trees [Database]

Given that: B = n/R blocks in the file 2d index records per block (blocking factor): 2d > R an extra block access from the index to the datafile I am not able ...
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### Red-Black tree height from CLRS

The lemma 13.1 of CLRS proves that the height of a red black tree with $n$ nodes is $$h(n) \leq 2\log_2(n+1)$$ There's a subtle step I don't understand. The property 4 reported at the beginning of ...
813 views

### How do I know which direction I should rotate a node in an AVL Tree?

I'm studying AVL Trees in my programming class and we got this exercise dealing with right, left, left-right and right-left rotations as a way to check if we understand the theoretical concept of AVL ...
1k views

### Red-Black Tree deletion algorithm (CLRS, 3rd edition) : Deleting the root

I have been following the third edition of Introduction to Algorithms (by Cormen, Rivest, et al.), and have been studying the deletion algorithm for red-black trees. However, I am confounded at the ...
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### 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 ...
3k views

### Can a Red Black tree be constructed of only black nodes using RB insert only?

I am trying to construct a red black tree out of only black nodes. I know it is possible getting it after some deletions but I am trying to construct one only via insertion orders. Is it possible? I ...
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### For a balanced binary search tree what is the worst case case time complexity for accessing all elements within a range of nodes?

I have this question which is asking for the worst case time complexity for a balanced binary search tree, assume the nodes are labeled as integers and we consider a range of ...
58 views

### Self-balancing BST supporting in-order-sequential multi-insertions / multi-deletions in logn+klogk time?

Given a self-balancing binary search tree of size $n$, I want to perform the following operations: InsertInOrderSequentialBatch an ordered sequence of $k$ values (...
1k views

### Is every AVL tree a BST or just BT?

I was going through the concepts of AVL tree and came across the definition of AVL tree from the wiki. In computer science, an AVL tree (named after inventors Adelson-Velsky and Landis) is a self-...
38 views

### Having trouble understanding Red-Black trees

Exam question: Draw the Red Black Tree that results from inserting the following values in the given order: [10, 20, 30, 4, 5, 50] Draw the red connections with a dotted line and the black ones with ...
408 views

### is it possible to create an avl tree given any set of numbers?

I am studying balanced trees, especially AVL trees. My question is whether is it possible to create an avl tree given any set of numbers. is it possible to prove the following statement? Let $A$ be ...
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### efficient DELETE in proto-Van Emde Boas Tree?

TLDR: CLRS is claiming that a certain "pseudo" or "proto" tree structure does not have fast deletion, but I seem to have an algorithm that is efficient, and I would like to know ...
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### Red-black tree amortized cost of the rebalancing

I've read in different sources that the amortized cost of a red-black tree rebalancing is constant (at least during the tree creation using only insertions). How can it be proven?
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### What is the maximal difference between the depths of 2 leaves in AVL tree?

I'm wondering what's the answer of the following question: What is the maximal difference between the depths of 2 leaves in an AVL tree? Intuitively I think that it shouldn't exceed $log n$ but have ...
840 views

### Worst Case for AVL Tree Balancing after Deletion

After deleting a node in an AVL tree, self-balancing (zig-zag rotation or the left-right balancing) maintains O(logn) time that is not guaranteed in other unbalanced trees (like BST). The Balancing ...
1k views

### What is the fastest way to merge two B trees?

Given two B-trees of some order $m$ - $T_1,T_2$, such that $y > x$ for every pair $x \in T_1$ and $y \in T_2$. What is the fastest way to create a new tree that is the union of both $T_1,T_2$? My ...
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### How to prove that in an AVL tree with height h, the depth of every leaf node is at least $\lceil h/2 \rceil$

I have an AVL tree with height h. I understand how to get h $\thickapprox$ 1.440 log N. However, I can't figure out how to calculate the minimum depth of a leaf node from root. I tried constructing a ...
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### What is the name of the data structure that is a tree on the backend but has a list like API?

I'm looking for the name of a data structure. It is organized like a balanced tree. The elements need not be comparable. Instead of asking if the tree contains a thing (like you would with a ...
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### Why do some search trees store all the elements in leaves, while others don't?

Why do some search trees store all elements in the leaves, while other search trees don't? One difference that makes is whether a successful search may end up in an internal node. For m-ary search ...
442 views

### Is "promoting a key" a part of deleting internal node key in B+ Tree?

I was trying to learn B+ tree deletion operation and trying to contrast it with B tree deletion operation. However barely any book provided detailed step by step B+ tree deletion operation. So I was ...
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### AVL tree worst case height proof

The worst case height of AVL tree is $1.44 \log n$. How do we prove that? I read somewhere about Fibonacci quicks but did not understand it.
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### Splitting a node in B+ tree with odd number of keys

This and several other resources suggest to "Always a node that gets the middle key from bottom splits, should drop one item for a new middle key". To illustrate with an example. For 5-way B+-tree, ...
84 views

### Scapegoat Trees: Why are they only loosely a-height-balanced?

From Wikipedia: Even a degenerate tree (linked list) satisfies this condition if α=1, whereas an α=0.5 would only match almost complete binary trees. A binary search tree that is α-weight-...
63 views

### Treap union unrandomizes priorities?

The algorithm for the union of two sets represented by treaps is defined in this paper and on Wikipedia, but the algorithm seems flawed. Take for example the following loop. ...
50 views

### What would happen if we added this rule to red-black trees?

So, I know that a normal r-b tree has a height of O(logn). What would happen is we let a red node have a red child if its parent is black? Would the height still be O(logn)? Would you have to have a ...
460 views

### How to find the number of intervals containing a point when given a static set of intervals?

I've seen similar questions around here but I'm trying to address this problem with a slight change and maybe it makes it easier to solve. I'm given a set of intervals $\{s_1,s_2,...,s_n\}$ where ...
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### Height of epsilon-balanced binary search tree

In Balanced Binary Search Trees on the basis of size of left and right child subtrees, Hannes says: For example, one can say, a BST is balanced, if each subtree has at most epsilon * n nodes, ...
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### 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 ...
67 views

### Linked lists with auxiliary data

I'm trying to design a data structure that supports the following operations in $O(1)$ time: $\operatorname{query}(\ell)$: Yield some auxiliary datum associated with list $\ell$ (just a single ...
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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 ...
197 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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### Don't understand one step for AVL tree height log n proof

I came across a proof that the an AVL tree has O(log n) height and there's one step which I do not understand. Let $N_h$ represent minimum number of nodes that can form an AVL tree of height h. Since ...
262 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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### Finding an interval in a binary search tree that contains a point

I have a binary search tree where nodes are non-overlapping intervals. I'm given a point, and I need to determine which interval the point belongs to (if any). This is easy to do because I can compare ...
400 views

### How to create an AVL tree, given it's pre-order traversal?

Given the pre-order traversal, we can easily obtain the necessary Binary Search Tree(BST- not necessarily balanced). However, the problem arises when we have to decide the balancing manually(this was ...
5k views

### AVL trees are not weight-balanced?

In a previous question there was a definition of weight balanced trees and a question regarding red-black trees. This question is to ask the same question, but for AVL trees. The question is, ...
230 views

### I don't understand the case 4 of red-black tree deletion

I don't know why case 4 will resolve the issue of the double black of $x$ described in Introduction to algorithm p.329. I know case 1 is transformed into one of {2,3,4} case, and case 2 re-point $x$ ...
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### Does keeping underflowed nodes unbalance the R-tree?

When deleting nodes from an R-tree, if a branch underflows, then it's meant to be dissolved and its children are meant to be reinserted into the tree. But most R-tree implementations that I've seen ...
306 views

### Counting chords intersections in a circle

The problem is: Given 2n distinct endpoints of n chords on the unit circle, count the number of intersections between chords (if k chords intersect at one point, that point counts as $\binom{n}{2}$ ...
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### Documentation of "bin number" trees

TL;DR: I implemented a special (?) binary tree and can't find any further details on the method I used on the internet. I would like to know if there are any scientific papers discussing my ...
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### Minimum finger search tree complexity

Suppose I have an AVL tree with a pointer to the minimal element. I'd like to conduct a search for some key x, which is the $k$-smallest key in the entire tree. I can do this by "climbing" up the tree'...
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### Efficient data structure for multidimensional searching on intervals and keys

I am searching for a data structure that can capture a database, which is consisted of one column of intervals (like [0, 2], [4, 6]) and one/two columns of keys (...
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### Why is b-tree search O(log n)?

B-tree is a data structure, which looks like this: If I want to look for some specific value in this structure, I need to go through several elements in root to find the right child-node. The I need ...
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### B/B+ trees without leftmost pointers

In both B-trees and B+trees, a node (a.k.a page) contains K keys and K+1 pointers: node = [ ptr_1, key_1, ... , ptr_K , key_K , ptr_(K+1) ] Now suppose that I ...
Is it true that inserting an element to an AVL tree requires $O(1)$ rotations? How many rotations, does deletion from AVL require? I've searched for these two questions with no luck so far.