Questions tagged [binary-trees]

a rooted tree in which each node has no more than two children

113 questions with no upvoted or accepted answers
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8
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0answers
617 views

Chained operations on sequences with two operators

Given a binary expresion tree, with addition and multiplication operations, how can we optimize it's evaluation? Can we learn from matrix chain multiplication? A generalization of matrix chain ...
6
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0answers
154 views

Impossibility of certain type of tree traversal algorithm

I was wondering for some time how to approach a situation like the following one. Imagine a standard binary tree data structure with $n$ nodes in it. Each node contains pointers to its left and right ...
6
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0answers
210 views

Correctness of a greedy Algorithm on Knockout Tournaments

You are given a function $\operatorname{rk}:\{1\dots 2^k\}\rightarrow \mathbb{N^+}$ representing the ranks of the players $1\dots2^k$ in a participating in a tournament. The tournament evolves in a ...
6
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2answers
274 views

Find if a sequence of node is a result of a binary tree preorder traversal

Assume we have a sequence of 0's and 1's: $n_0, n_1, ..., n_N$, in which 0 stands for a leaf node, and 1 stands for an uncertain node (it may or may not be a leaf node). How to check if this sequence ...
5
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240 views

Average redundancy in Huffman or Hu-Tucker codes on random symbol probabilities

Huffman and Hu-Tucker codes are well-known compression schemes, which both come close to the entropy lower bound. It is known that if $L_1$ and $L_2$ are the lengths of a Huffman resp. Hu-Tucker code, ...
5
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0answers
323 views

Efficient deletion in Fenwick tree

Suppose there is an array of nonzero integer values A[n], which has a Fenwick tree representation F[n]. The most simplistic way ...
4
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0answers
93 views

Number of binary trees of size $n$ such that all subtrees of same size are equal?

In the following, I consider rooted, unlabelled, ordered binary trees, where each node has exactly $0$ or $2$ children (I will simply call them binary trees). A binary tree $t'$ is a subtree of a ...
4
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1answer
649 views

Optimal Binary Search Trees Knuth

Knuth, Donald E. (1971), "Optimum binary search trees", Acta Informatica 1 (1): 14–25,doi:10.1007/BF00264289 Please have a look at this paper, specifically page 18 in which he tries to prove his ...
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0answers
45 views

How to find the Expected height of a randomly built binary tree

I would like to find out the Expected height of a binary tree where the insertions are based on a random function. I.e. for each node I visit, there is a $\frac{1}{2}$ probability of choosing right or ...
3
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0answers
64 views

Node depth in randomly built binary search tree

It can be proved that randomly built binary search trees of size $n$ are of depth $O(\log n)$ and it is clear that level $k$ has at most $2^{k}$ nodes (root's level is 0). I have an algorithm that ...
3
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0answers
335 views

What's the number of leaves in AVL tree

How can I prove that the number of leaves in a balanced BST is $\Omega (N)$ where $N$ is the number of nodes in the tree? I tried somehow to prove that an AVL/Fibonacci tree should have $\Omega (N)$ ...
3
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42 views

Proving an upper-bound on the cost of all calls to rebuild() during a sequence of m operations in a scapegoat tree

I'm reading through Open Data Structures by Pat Morin and am currently looking at scapegoat trees. The book is free and can be downloaded from here: http://opendatastructures.org/ods-java.pdf On page ...
3
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0answers
147 views

Total number of red black tree arrangements

Consider a list of n integers. And assume we take each integer from this list sequentially and add it to a red-black tree. Then for the n! permutations of the list how many red-black tree ...
3
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0answers
297 views

What is the intuition behind balancing in AVL trees?

I am not sure that my question is clear from the first sight. But I will try to explain what I mean. For now, I am learning balancing the trees on the example of AVL trees. We know that to balance ...
3
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0answers
558 views

Range bit inversions and range set bit queries with a binary indexed tree

I am trying to solve this problem using a binary indexed tree. The problem can be summarized as follows: You are given a series of commands that operate on an array initially all zeroes. 1) ...
3
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1answer
945 views

what is the name for the space between the leaves of a tree

I am trying to write a data-type not for a tree, but for the spaces in between the leaves of thee tree. In number theory (a part of math) this is known as a topograph does it have a name in CS?
2
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0answers
19 views

Repairing a perfect binary merkle forest

Let's say we have a forest of imperfect binary merkle trees of different heights (this is an invariant). Only the numbered nodes store data, the rest store hashes. Here is an example with 4 trees, ...
2
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0answers
23 views

How to linearly combine loss functions to preserve optimal substructure property?

I've been working on a binary tree optimization problem with two choices of loss function (let's call them A and B). I'm fairly certain that the problem of minimizing either A or B individually has ...
2
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0answers
24 views

Data Structures for BST where size uniquely determines shape

There are several data structures in which the number of elements uniquely determines the shape. Examples would be binary heaps, arrays, lists, Braun trees, and Merkle mountain ranges. Are there any ...
2
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0answers
26 views

Nested dissection vs kd-tree

Could you explain, please, the difference between the nested dissection and kd-tree. For me they look same representing a tree data structure for a distribution of points in a multi-dimensional ...
2
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0answers
40 views

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 ...
2
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0answers
420 views

Analysis expected depth of a binary search tree given random values?

I have a guess about the problem above. Suppose I have a binary search tree $T$ initially empty. Suppose I drawn $x_1,\ldots,x_k$ (from some real interval $[a,b]$) keys and I want to insert the keys ...
2
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1answer
220 views

Splay trees: why are depths of nodes on the access path halved?

The original paper describing splay trees Self-Adjusting Binary Search Trees by Sleator and Tarjan claims that: Splaying not only moves x to the root, but roughly halves the depth of every node ...
2
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148 views

Algorithm to find nodes with given a distance

We have a map of a branching river without any islands (tree of confluences, see picture), there are piers on the river bank ( not necessarily on the confluences). We are given the distances between ...
2
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0answers
705 views

Tight upper bound on height of a red-black tree

"Introduction to Algorithms" by Cormen et al., 3rd edition, Lemma 13.1 states that A red-black tree with $n$ internal nodes has height at most $2\log(n+1)$, i.e. $h \le 2\log(n+1)$. Can equality ...
2
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0answers
141 views

Finding post-order traversal of a binary tree from its in-order and pre-order traversals lower bound

I know that we can construct a BST by just having its pre-order traversal in $O(n)$ time (this link). But what if the tree is just a binary tree and we have its in-order and pre-order traversals? I ...
2
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0answers
115 views

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 ...
2
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0answers
251 views

WAVL Trees - Number of rebalances

I was experimenting with a WAVL Tree code I wrote. One thing I noticed is that on average, there was much less rebalancing actions after delete than after insert. In fact, average rebalances after ...
2
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0answers
102 views

How to cut a full binary tree evenly

Given a full binary tree of size $n$, I need to cut the tree into two by choosing an edge and removing it. I want the cutting to be as equal as possible (i.e. the two subtrees have as equal weight as ...
2
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1answer
261 views

The Height of subtree in Heap

In order to find the recurrence function of The Height in Heap, the following figure is drawn. Question 1: How can we compute the height if Right subtree in form of Log in base 3, and why do we have ...
2
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0answers
19 views

How to avoid empty spots when inserting in a B+tree

When you insert a new item in a B+tree leaf node, and the node is full, two new nodes are created and they're both filled halfway and the values get re-distributed accordingly. Before the insertion: ...
2
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1answer
676 views

How to efficiently create balanced KD-Trees from a static set of points

From Wikipedia, KD-Trees: Alternative algorithms for building a balanced k-d tree presort the data prior to building the tree. They then maintain the order of the presort during tree construction ...
2
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0answers
654 views

Red black tree - Insert/Delete proof of correctness

From the Cormen book I was studying the chapter focused on the red black tree. I was particularly interested in why the procedures for insert/delete fixup works (namely a formal proof). I report both ...
2
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0answers
83 views

Can the right childs of a node in an AVL tree be both balanced after an insert and need rebalancing?

I was trying to come up with a case where one would need rebalance the following case in an AVL tree: I think that case impossible to happen during an insert. It seems to me that its impossible to ...
2
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0answers
55 views

Two flavors of red-black trees with different performance

I've implemented two red-black trees (using the pseudo-code in CLRS) with slightly different flavors: 1) In the first tree, all the data is stored in the leaves. 2) In the second tree, the data is ...
2
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0answers
276 views

Marking nodes of a complete binary tree

Suppose that I have a binary tree with $N = 2^h - 1$ nodes, initially all nodes are unmarked Over time via this process nodes became marked. Suppose that nodes have unique identifiers in range of $[1,...
2
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0answers
41 views

How to find number of occurences of specific distances in binary (search) trees?

I want to calculate the amount of tree structures that have a given maximal distance between two nodes given an amount n of nodes (or keys). E.g. with ...
2
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0answers
241 views

Difference between fully-reduced BDD and quasi-reduced BDD

I am trying to figure out difference between fully- and quasi-reduced BDDs. I have read a lot of material but still it is not very clear. As I am trying to figure out the quasi reduced version for ...
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0answers
30 views

Weight of lowest common ancestor satisfies strong triangle inequality

How do I prove that $d(x,y)$, defined as the weight of the lowest common ancestor of $x,y$, satisfies the strong triangle inequality: $$ d(x,y) \le \max(d(x,z), d(y,z)) $$ How do I even start such a ...
1
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1answer
40 views

Height of AVL tree with balance condition of 2

The maximum height of an AVL tree with a balance condition of 1 is 1.44log(n). So the worst case height is O(logn). However, if the balance condition was hypothetically 2 (meaning that the allowed ...
1
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1answer
744 views

Why does the formula floor((i-1)/2) find the parent node in a binary heap?

I learned that when you have a binary heap represented as a vector / list / array with indicies [0, 1, 2, 3, 4, 5, 6, 7, 8, ...] the index of the parent of element at index i can be found with parent ...
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0answers
24 views

Recording a histogram in a tree exhibits strange best case

The task is to record a histogram from a streaming data source. One data point is, say, a 16 bit integer. The maximum multiple of one data point before the stream ends, is < 2^32. The main ...
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0answers
862 views

Find the median element of two AVL trees in $O(\log n)$

I'm attempting the problem of finding the median element in two AVL BST's in $O(\log n)$ time. In this problem, we are given two AVLs, with a combined size of $n$ (the distribution across the two ...
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0answers
24 views

Determining Range of Current Node in Segment Tree

I was attempting, though failing quite miserably, to find some method of of determining the range of some node $n$. By range I mean an interval $[l,r]$ over the base array that is reachable by the sub-...
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0answers
13 views

Confusion with “every path from a given node to any of the leaves goes through the same number of black nodes” property of RB trees

One of the properties of Red Black trees is: "every path from a given node/vertex to any of the leaves goes through the same number of black nodes" Two related questions about this property: 1) is ...
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0answers
33 views

relationship between binary numbers and binomial heaps

I understand that a binomial heap can be represented as binary numbers according to the degree of each tree but what exactly is the relationship between inserting a new node into the binomial heap and ...
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0answers
446 views

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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0answers
38 views

Finding a node in a binary tree by looking at the path between it and the root

There is a directed binary tree as shown in the picture (all edges are diercted from higher- to lower-level nodes). In that tree there is some specific unknown node $s$. All nodes in the $(s, root)$ ...
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0answers
67 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-...
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0answers
62 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 (...