28
votes
Accepted
Can the pre-order traversal of two different trees be the same even though they are different?
Tree Examples (image):
...
- 576
17
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 ...
- 369
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 ...
- 3,068
9
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 ...
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 ...
- 275k
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 ...
- 28.5k
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
$...
- 275k
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 $...
- 29.9k
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.♦
- 156k
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,
...
- 72.1k
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 ...
- 4,448
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.♦
- 156k
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 ...
- 28.4k
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 ...
- 21.7k
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 ...
- 9,501
6
votes
If both could be implemented with the other, what are the differences between priority queues and binary heaps?
Based on standard usage of the terms, a heap is a specific data structure, with a specific representation in memory. A priority queue is an abstract data type: it identifies some operations that must ...
D.W.♦
- 156k
6
votes
Accepted
Using pre-,post-, and in-order indexes to find information about a Binary Search Tree
Long story short: it is possible in constant time if the tree is a full binary tree. If not, there are some cases where there is not enough information to find the size of the subtree in constant time....
- 12.5k
6
votes
Accepted
An α-good tree with n nodes has height O(log n)
$$2|y| = (|y|-|z|) + (|y|+|z|)\le \alpha |x| + |x| - 1.$$
So, $|x| \ge \frac2{1+\alpha}|y|$.
Since $y$ is an arbitrary child of $x$, if node $x$ is of height $k$, $|x| \ge \left(\frac2{1+\alpha}\right)...
- 38.6k
6
votes
Accepted
Are reversed reverse preorder traversals equivalent to a postorder traversal?
This can be proven by induction on trees. I give details on the conjecture 1 here.
It is clearly true for the empty tree and for leaves;
Suppose it is true for trees $l$ and $r$. Consider $t$ a node ...
- 12.5k
5
votes
Find the longest possible path in full binary tree
If it is a full binary tree, that is defined as:
Full binary tree is a tree in which every node other than the leaves has two children.
Then you know the depth $D$ will be half of the total ...
- 4,431
5
votes
Accepted
Every AVL tree may be red black tree
Your proof produces a tree in which all nodes are colored black. It doesn't necessarily satisfy the "black height" rule:
Every path from a given node to any of its descendant NIL nodes contains the ...
- 275k
5
votes
Depth first or breadth first ordering in binary search trees?
Think about what happens when you move from one layer in the tree to the next. When you start getting to layers with progressively more nodes, you'll eventually get to a spot where the layers are so ...
- 9,037
5
votes
Help with proof involving weighted full binary tree
There are two basic induction patterns for (non-empty) full binary trees:
A tree is either a leaf or consists of a root and two full binary subtrees.
A tree is either a leaf or can be obtained from a ...
- 275k
5
votes
Accepted
Depth first or breadth first ordering in binary search trees?
There's a paper on this: Khuong and Morin. Array Layouts For Comparison-Based Searching
They compare the Eytzinger, B-Tree, Van Emde Boas, and sorted array layouts and conclude that Eytzinger works ...
- 1,274
5
votes
Accepted
Is a balanced binary tree a complete binary tree?
A complete binary tree is a binary tree of length $h$ such that all the levels from $1$ to $h-1$ are completed and the last level gets completed from left to right. As in the image below.
A balanced ...
- 156
5
votes
Accepted
How many number of different binary trees are possible for a given postorder (or preorder) traversal
Every binary tree (with the right number of nodes) has exactly one labelling that satisfies a given postorder labelling. So you need to find the number of binary trees. That is the famous Catalan ...
- 29.9k
5
votes
Accepted
KD-Tree implementation with lat/lon coordinates
Short answer: No, you will not get the same results.
It does not matter what coordinate system or what geographic standard you are using, it is not possible to make projection from sphere into ...
- 9,425
5
votes
Accepted
Why is this not a valid Red-Black tree?
If you go to the empty leaf from the root in the pattern [Right, Left], you get to an empty leaf encountering 1 black node. If you go [Right, Right, Left] or [Right, Right, Right], you get to an empty ...
- 66
5
votes
Binary Search Tree Traversals
The rules for traversing a tree are, to visit all nodes of a tree you do this:
Preorder: (1) visit the root node, then (2) visit all the nodes in the left subtree of the root, then (3) visit all the ...
- 14.8k
Only top scored, non community-wiki answers of a minimum length are eligible