Questions tagged [binary-trees]
a tree in which each node has no more than two children
380
questions
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0answers
30 views
If insertion/deletion from a binary tree is efficient, while maintaining ability to get item by index
I can't quite figure this out. Say you have a binary tree where the left or right nodes correspond to 0 or 1 and a group of levels form a chain which is the index of the node. So you have 10010 which ...
2
votes
2answers
26 views
Randomly built binary search trees
In Introduction to Algorithms (CLRS) 3rd Edition, page 299, the section attempts to prove:
The expected height of a randomly built binary search tree on $n$ distinct keys is $O(\lg n)$.
We define "...
4
votes
1answer
46 views
Nodes in a binary search tree that span a range
I have a binary search tree of height $h$ with an integer in each leaf. I also have a range $[\ell,u]$. I want to find a set of nodes that span the range $[\ell,u]$, i.e., a set $S$ of nodes such ...
3
votes
1answer
59 views
Finding $k$-th element in prefix of size $i$
Let's say we are given array $A$ of size $n$. We need to answer some numbers of queries. For each query we are given index $i$ and integer value $k$, $k \le i$. If we take the first $i$ elements of ...
11
votes
4answers
3k views
Can the pre-order traversal of two different trees be the same even though they are different?
This question pretty much explains that they can, but does not show any examples of there being two different trees with the same pre-order traversal.
It is also mentioned that the in-order ...
2
votes
3answers
374 views
Do height-balanced binary trees have logarithmic depth?
A family of binary trees is called balanced if for every tree $t$ in the family the height of $t$ is $O( \log n)$.
Given a family of trees such that for every tree $t$ in the family, for every node $...
2
votes
0answers
17 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
votes
2answers
102 views
Average codeword length in a Huffman tree is $\Omega(\log n)$
Prove that the average codeword length in a Huffman tree is $\Omega(\log n)$, where $n$ is the number of characters.
My try:
I think that the worst case is when the tree is full and all the ...
1
vote
1answer
25 views
Number of nil-links in a binary tree
In its section Properties of binary trees Wikipedia states:
The maximum possible number of null links (i.e., absent children of the nodes) in a complete binary tree of n nodes is (n+1), where only ...
0
votes
0answers
19 views
Do parent pointers affect path-copying in a persistent binary tree?
I am familiar with the concept of path-copying in the context of binary trees to achieve persistent modifications, but I am unsure if the inclusion of parent pointers has any effect on the complexity. ...
1
vote
1answer
69 views
Finding longest subset arithmetic progression with given difference
Given a list of distinct positive integers, I am trying to find the largest subset that forms an arithmetic sequence with a given difference D.
For example, given D = 5, with the set of numbers 1, 5, ...
1
vote
1answer
36 views
Self-balancing binary search tree optimized for insertion
I've written a "quiz" that prompts the user for comparisons between two items of subjective value, and once the position of all of the items is determined, displays an ordered list from most valuable ...
0
votes
1answer
71 views
Bin packing first-fit problem in $O(n \log n)$ time
Suppose we have $n$ objects with weights $w_i \in (0,1]$ and we must
insert them into bins with the constraint that every bin must contain
objects which weight less than $1 \, kg$.
The first-...
1
vote
3answers
81 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?...
0
votes
0answers
22 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 ...
2
votes
1answer
40 views
Logic of multiple variables in ILP
Is there a better way to represent an AND of $n$ variables together other than creating $O(n)$ new variables and constraints?
1
vote
2answers
83 views
Difficulty understanding the solution of heap problem in CLRS book?
I am reading the solution of this problem in CLRS:
Show that there are at most $\lceil {n/2^{h+1}} \rceil$ nodes of height $h$ in any $n$-element heap.
Solution: All the nodes of height $h$ ...
1
vote
2answers
102 views
Random walks on Complete Binary Trees
Let $T$ be a complete binary tree of height $n$ and root $r$.
A random walk starts at $r$, and at each step uniformly at random moves on a neighbor.
There are $m$ random walkers all starting at $r$ ...
0
votes
0answers
75 views
Binary search on a path of minimum heap
WhereTo(H,X) is searching for the place to set X (an integer) in a minimum heap-H.
The function is executing a binary search on a path of a heap.
Assumption: We have the specific path because it ...
1
vote
1answer
214 views
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.
1
vote
2answers
63 views
Build a balanced binary tree from list in linear time
I couldn't figure this one out. Given an unsorted list, we want to build a balanced binary tree (not a search tree, namely - left child is of lower key, and right child of higher key). Can we do it in ...
0
votes
0answers
28 views
How does a binary tree waste memory when stored as nodes and references?
I'm researching binary trees and came across this section describing storage methods.
It states that:
In a language with records and references, binary trees are typically constructed by having a ...
2
votes
1answer
56 views
Merging of two increasing binary trees why do we exchange left and right subtree
We say that a binary tree : $B(l, x, r)$ is increasing if $x$ is smaller than all the nodes of the binary tree and if $l$ and $r$ are also increasing trees.
One way to merge these kind of trees is as ...
1
vote
2answers
48 views
Minimal, “natural” set of operations to manipulate a binary tree
Consider binary trees with some abstract values at the leaves. A tree t2 is said to be "derivable" from a tree t1 iff the set of values in the leaves of t2 is included in the set of values in the ...
0
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0answers
43 views
Storing binary tree with arbitrarily sized nodes without linked-list or large 32-bit pointers
So you can store a binary tree without pointers using a 1-D array:
Binary Trees can be represented using 1-D array in memory(Fig 1).The rule to store binary tree in array are :
The values of ...
2
votes
1answer
61 views
Prove number of nodes in heap
Consider a variation of the normal heap which we will call the x-heap
The x-heap of height $h$ has the following properties:
It will have $2^h$ nodes
A height of $0$ corresponds to the single root ...
1
vote
1answer
224 views
$O(n\log n)$ algorithm for minimizing number of inversions in leaves of complete binary tree
I'm having trouble making an algorithm to fit these specs:
Given a complete binary tree ($n = 2^d$ leaves) with integers in leaves.
Reading the leaves from left to right makes a sequence of integers (...
2
votes
1answer
27 views
Binary Tree Multithreaded Time Complexity
Let's say you want to get the value of all nodes in a binary tree (order doesn't really matter). If in each thread, you spawn two more threads to deal with the left subtree and the right subtree, then ...
2
votes
1answer
52 views
Algorithm to find the two closest elements in a BST
For example, consider an AVL tree:
Performing the two_close(T) should give back (13, 12), as they are the closest elements together. Said another way, you can take any two other elements in the tree, ...
1
vote
1answer
74 views
Finding optimal multiplication order and optimal binary tree
I have to determine the Optimal Multiplication Order for above matrices using Dynamic Programming approach and also present that sequence (i.e. optimal order) in Binary Tree.
Consider the following ...
0
votes
0answers
41 views
Binary tree for 2 elements
I want to understand Binary Search for 2 element list made of 1,2. I draw a tree as below. Is it correct?
If I want to search for an element 2, it will make 2 comparisons. If I want to search for an ...
2
votes
0answers
33 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 ...
5
votes
2answers
268 views
What is the point of traversing a binary tree in preoder, inorder or postorder?
Why would you want to traverse a binary tree in preoder, inorder or postorder? Why not use an order like breadth-first search for all graphs?
1
vote
1answer
41 views
AVL Rotations Abbreviations
I read that there are 4 types of rotations:
Left rotation
Right rotation
Left-Right rotation
Right-Left rotation
What are the corresponding abbreviations: RL,LR,RR,LL? Is there a reference how to ...
1
vote
1answer
56 views
Optimize sorting matrix entries by row and column
I am writing a routine to store an $M$-by-$N$ sparse matrix in a balanced binary tree. The insertion routine calls a comparison function to determine where a new matrix entry $(i,j)$ should be ...
1
vote
1answer
34 views
Binary trees and preallocated nodes
I want to design a binary tree with preallocated nodes, in order to avoid calling malloc/free every time I want to insert/delete a node. The problem is I don't know ahead of time how many nodes the ...
1
vote
1answer
268 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 ...
1
vote
1answer
54 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 ...
1
vote
1answer
25 views
Running time based on smallest subtree
I've constructed an algorithm on (rooted) binary trees, where the running time at a node depends on the size of its smaller subtree, where we compute from each leaf upwards towards the root.
More ...
1
vote
2answers
101 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 ...
0
votes
1answer
29 views
How can we form a graph from this problem?
So from the above diagram, we can move either right or down. So a binary tree is built that way. It's written ...
2
votes
1answer
384 views
The number of balanced trees with N node and L leaves
An algorithm is requested to calculate all balanced binary trees which can be built with $N$ nodes, having exactly $L$ leaves. A balanced tree is a binary tree in which the difference between the ...
1
vote
2answers
62 views
is a deterministic algorithm for breadth-first traversal possible (binary heap)
Imagine I have a binary heap, traversed in a breadth first manner like so:
...
2
votes
2answers
1k views
Proving Postorder Traversal's Time Complexity
I am looking at the following algorithm for performing a Postorder Traversal of a binary tree
...
1
vote
2answers
343 views
Why can't we just use preorder traversal to check if a tree is subtree of binary tree?
Is preorder traversal enough to check if a tree is subtree of a binary tree?
Are there any scenarios which I can miss if I use just the preorder traversal?
What other methods can be used to check if ...
2
votes
1answer
605 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 ...
2
votes
1answer
65 views
Selecting a subtree in an array representation of a binary tree
Consider a zero-indexed array representation of a binary tree (as in a binary heap), where
0 is the root
The left child of i is ...
-4
votes
1answer
36 views
Construct an array from a binary tree
I am looking for a suggestion. I know it is possible to construct a binary tree from an array. I want to know that vice verse is true? Is it possible to construct an array from a binary tree? If yes ...
1
vote
0answers
54 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 (...
4
votes
2answers
2k views
Time Complexity to find height of a BST
Below is a question I was asked in an Interview
What is the best case time complexity to find the height of a Binary Search Tree?
I answered it explaining using the below algorithm
$\mathrm{...
