91
votes
Accepted
What's the difference between a binary search tree and a binary heap?
Heap just guarantees that elements on higher levels are greater (for max-heap) or smaller (for min-heap) than elements on lower levels, whereas BST guarantees order (from "left" to "right"). If you ...
- 1,092
49
votes
37
votes
What's the difference between a binary search tree and a binary heap?
Both binary search trees and binary heaps are tree-based data structures.
Heaps require the nodes to have a priority over their children. In a max heap, each node's children must be less than itself. ...
- 1,711
28
votes
Accepted
Can the pre-order traversal of two different trees be the same even though they are different?
Tree Examples (image):
...
- 566
20
votes
Accepted
What are the main ideas used in a Fenwick tree?
A Fenwick tree is a binary tree used to efficiently handle cumulative frequencies or sums in an array.
Without loss of generality we shall examine a 16-element array. Imagine a binary tree imposed on ...
- 381
18
votes
Accepted
Does a graph always have a minimum spanning tree that is binary?
There is nothing to be done: for instance, let $S_k$ denote the star graph with $k$ leaves. The graph $S_k$ has a unique spanning tree (which is $S_k$ itself), and it has a vertex with degree exactly $...
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14
votes
Accepted
When are binary trees better than hashtables in real world applications?
Hash tables can only tell you if an element is present or not.
Here are somethings you can do with a binary tree that you can't do wiht a hash table.
sorted traversal of the tree
find the next ...
- 522
14
votes
BIT: What is the intuition behind a binary indexed tree and how was it thought about?
I think that the original paper by Fenwick is much clearer.
The answer above by @templatetypedef requires some "very cool observations" about the indexing of a perfect binary tree, which are ...
- 339
13
votes
What's the difference between a binary search tree and a binary heap?
With data structure one has to distinguish levels of concern.
The abstract data structures (objects stored, their operations) in this question are different. One implements a priority queue, the ...
- 27.6k
12
votes
Size of decision tree and depth of decision tree
The depth of a decision tree is the length of the longest path from a root to a leaf.
The size of a decision tree is the number of nodes in the tree.
Note that if each node of the decision tree ...
D.W.♦
- 141k
12
votes
Accepted
What is "rank" in a binary search tree and how can it be useful?
According to this book (Chapter 3.2), a node in a BST has rank $k$ if precisely $k$ other keys in the BST are smaller. So, if you order all the BST nodes according to their keys, then each node with ...
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11
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When are binary trees better than hashtables in real world applications?
One application domain where binary trees are better, or more easily adjustable than certain alternatives, are persistent data structures (which are often used in (purely) functional programming).
A ...
- 667
9
votes
Accepted
Range update + range query with binary indexed trees
Suppose you had an empty array:
0 0 0 0 0 0 0 0 0 0 (array)
0 0 0 0 0 0 0 0 0 0 (cumulative sums)
And you wanted to make a range update of +5 ...
- 406
9
votes
Time Complexity to find height of a BST
Your algorithm runs in linear time on all inputs. The algorithm visits each node of the tree exactly once, and does $O(1)$ work per node. Therefore it runs in time $\Theta(n)$, where $n$ is the number ...
- 270k
9
votes
Can the pre-order traversal of two different trees be the same even though they are different?
Counting argument
The number of unlabeled binary trees of $n$ nodes is the $n^\text{th}$ Catalan number $C_n=(2n)!/(n!(n+1)!).$ For example there are 5 binary trees of 3 nodes,
...
- 366
9
votes
Find the number using binary search against one possible lie
A generalization of this class of problems is widely studied. See, e.g., this paper for a survey.
In your particular case, the problem can be easily solved without any asymptotic change in the ...
- 23.4k
8
votes
Accepted
Lower bound of a summation with an exponential
There is a general formula for this sum:
$$
\sum_{h=0}^m h2^h = \sum_{h=1}^m \sum_{k=1}^h 2^h = \sum_{k=1}^m \sum_{h=k}^m 2^h = \sum_{k=1}^m (2^{m+1}-2^k) = m2^{m+1} - (2^{m+1}-2).
$$
Overall, we get
$...
- 270k
8
votes
Accepted
How are binary trees represented on disk
This may not be an exact answer but some information of interest related to your question
Other answers have mentioned various ways in which the binary data structure can be represented and you might ...
8
votes
Can the pre-order traversal of two different trees be the same even though they are different?
Lets assume you consider trees of $n$ nodes. Now take any binary tree with $n$ nodes and name the nodes according to their pre-order numbering. Then clearly the pre-order sequence of the tree will be $...
- 27.6k
7
votes
Accepted
Number of Different AVL Tree
Let $a_{n,h}$ denote the number of AVL trees with $n$ nodes and height $h$. It is straightforward to get a recurrence for $a_{n,h}$:
$$a_{n,h} = \sum_{k=1}^n \bigl(a_{k-1,h-1}a_{n-k,h-1} + a_{k-1,h-1}...
- 6,519
7
votes
Accepted
When inserting into a binary tree, is there a universal agreed upon place to insert then new node to minimize complexity?
There are several different concepts here. Let's start from the most general. A tree is a data structure that consists of a root and a collection of children, each of which is a tree. A node of the ...
7
votes
Have I invented a new data structure?
I've never seen this data structure before. However, it doesn't seem like a good choice for storing a set of words, for most purposes. I see three significant disadvantages:
Performance. Looking up ...
D.W.♦
- 141k
7
votes
How are binary trees represented on disk
There are many ways to represent trees, each with their own set of advantages and disadvantages.
Here's an incomplete list:
Use any graph representation, e.g.
adjacency matrix,
incidence matrix,
...
- 70.9k
7
votes
What is the point of traversing a binary tree in preoder, inorder or postorder?
Different traversals of a binary tree exist to suffice different data dependencies between the nodes.
Let's have a comparison between different traversals of a tree. Note that aside from in-fix ...
- 4,083
7
votes
Can the same node appear twice in a tree?
A tree is defined to be a set of nodes, with a parent-child relationship that satisfies certain properties. Thus, it doesn't make sense to ask whether a node can "appear" twice.
In your code snippet,...
D.W.♦
- 141k
7
votes
Find the number using binary search against one possible lie
If normal binary search would take k questions, then you can solve this with 2k+1 questions: Ask each question twice. If you get the same answer, it was the truth. If not, a third question reveals the ...
- 25.2k
6
votes
Accepted
Check whether it is possible to turn one BST into another using only right-rotations
If $T_1$ and $T_2$ don't encode the same sequence, they are not right-convertible into each other (obviously). This can be tested in linear time by inorder traversal of both trees.
The following ...
- 6,519
6
votes
Why aren't tries generally used?
This is an interesting question. Certainly worth asking.
The choice of a data structure is very much dependent on what you want
to do with it. A more costly sophisticated structure, no matter how
...
- 19.1k
6
votes
Accepted
Huffman Code VS Hu–Tucker Code
Let me give you a real-world example, that's very similar to something I wrote once.
Let's say you're implementing a library catalogue system. A library catalogue is conceptually a collection of ...
- 18.9k
6
votes
Accepted
Maximal difference of height between two leaves in an AVL tree
We consider Fibonacci tree ([TAOCP3, Knuth98, Sect. 6.2.1]) and compute the maximal height difference in it.
A Fibonacci tree of order $k$ which is constructed recursively (see an Fibonacci tree of ...
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