I know what Binary search tree is and I know how they work. But what does it take for it to become a skewed tree? What I mean is, do all nodes have to go on one side? or is there any other combination?
Having a tree in this shape (see below) is the only way to make it a skewed tree? If not, what are other possible skewed trees?
Skewed tree example:
Also, I searched but couldn't find a good solid definition of a skewed tree. Does anyone have a good definition?
Figured out a skewed Tree is the worst case of a tree.
`
The number of permutations of 1, 2, ... n = n!
The number of BST Shapes: (1/n+1)(2n!/n!n!)
The number of skewed trees of 1, 2, ....n = 2^(n-1)
`
Here is an example I was shown:
http://i61.tinypic.com/4gji9u.png
A good definition for a skew tree is a binary tree such that all the nodes except one have one and only one child. (The remaining node has no children.) Another good definition is a binary tree of n nodes such that its depth is n-1.
A binary tree, which is dominated solely by left child nodes or right child nodes, is called a skewed binary tree, more specifically left skewed binary tree, or right skewed binary tree.
Related
I know what Binary search tree is and I know how they work. But what does it take for it to become a skewed tree? What I mean is, do all nodes have to go on one side? or is there any other combination?
Having a tree in this shape (see below) is the only way to make it a skewed tree? If not, what are other possible skewed trees?
Skewed tree example:
Also, I searched but couldn't find a good solid definition of a skewed tree. Does anyone have a good definition?
Figured out a skewed Tree is the worst case of a tree.
`
The number of permutations of 1, 2, ... n = n!
The number of BST Shapes: (1/n+1)(2n!/n!n!)
The number of skewed trees of 1, 2, ....n = 2^(n-1)
`
Here is an example I was shown:
http://i61.tinypic.com/4gji9u.png
A good definition for a skew tree is a binary tree such that all the nodes except one have one and only one child. (The remaining node has no children.) Another good definition is a binary tree of n nodes such that its depth is n-1.
A binary tree, which is dominated solely by left child nodes or right child nodes, is called a skewed binary tree, more specifically left skewed binary tree, or right skewed binary tree.
I am a beginner in the field of data structures, I am studying binary trees and in my textbook there's a tree which is not a binary tree but I am not able to make out why the tree is not a binary tree because every node in the tree has atmost two children.
According to Wikipedia definition of binary tree is "In computer science, a binary tree is a treedata structure in which each node has at most two children, which are referred to as the left child and the right child."
The tree in the picture seems to satisfy the condition as mentioned in the definition of binary tree.
I want an explanation for why the tree is not a binary tree?
This is not even a tree, let alone binary tree. Node I has two parents which violates the tree property.
I got the answer, This not even a tree because a tree is connected acyclic graph also a binary tree is a finite set of elements that is either empty or is partitioned into three disjoint subsets. The first subset contains a single element called the root of the tree. The other two subsets are themselves binary trees called the left and right subtrees of the original tree.
Here the word disjoint answers the problem.
It's not a binary tree because of node I
This can be ABEI or ACFI
This would mean the node can be represented by 2 binary numbers which is incorrect
Each node has either 0 or 1 parents. 0 in the case of the root node. 1 otherwise. I has 2 parents E and F
What's wrong with the heights of the left and right children of some node differ being by 2?
This is my first encounter with AVL trees, and I can't seem to understand why it is a must?
Really, what's wrong with the children being different by 2?
Regards
By definition, a balanced binary tree can only differ by one. If you look at the algorithms that operate on the AVL trees, you'll see that this property is always maintained.
While it's possible to make some sort of data structure where the height differs by +- 2 at most, there is not real benefit in doing so. By leaving it as +- 1, you create a simpler, self-balancing data structure.
Well that's the concept of an AVL tree, the height of left and right childs must not differ by at most one.
From Wikipedia
In an AVL tree, the heights of the two child subtrees of any node
differ by at most one.
Since it's balanced, this make the search 0(logn), so it's faster than a non-balanced binary tree, where all the elements could be on the left, making it 0(n)
There are lots of questions around about red-black trees but none of them answer how they work. Why is it called red-black? How does this keep the tree balanced (thus increasing performance over an unbalanced normal binary search tree)? I'm just looking for an overview of how and why it works.
For searches and traversals, it's the same as any binary tree.
For inserts and deletes, more sophisticated algorithms are applied which aim to ensure that the tree cannot be too unbalanced. These guarantee that all single-item operations will always run in at worst O(log n) time, whereas in a simple binary tree the binary tree can become so unbalanced that it's effectively a linked list, giving O(n) worst case performance for each single-item operation.
The basic idea of the red-black tree is to imitate a B-tree with up to 3 keys and 4 children per node. B-trees (or variations such as B+ trees) are mainly used for database indexes and for data stored on hard disk.
Each binary tree node has a "colour" - red or black. Each black node is, in the B-tree analogy, the subtree root for the subtree that fits within that B-tree node. If this node has red children, they are also considered part of the same B-tree node. So it is possible (though not done in practice) to convert a red-black tree to a B-tree and back, with (most) structure preserved. The only possible anomoly is that when a B-tree node has two keys and three children, you have a choice of which key to goes in the black node in the equivalent red-black tree.
For example, with red-black trees, every line from root to leaf has the same number of black nodes. This rule is derived from the B-tree rule that all leaf nodes are at the same depth.
Although this is the basic idea from which red-black trees are derived, the algorithms used in practice for inserts and deletes are modified to enforce all the B-tree rules (there might be a minor exception - I forget) during updates, but are tailored for the binary tree form. This means that doing a red-black tree insert or delete may give a different structure for the result than that you'd expect comparing with doing the B-tree insert or delete.
For more detail, follow the Wikipedia link that MigDus already supplied.
A red-black tree is an ordered binary tree where each vertex is coloured red or black. The intuition is that a red vertex should be seen as being at the same height as its parent (i.e., an edge to a red vertex is thought of as "horizontal" rather than "descending").
[I don't believe the Wikipedia entry makes this point clear.]
The usual rules for red-black trees require that a red vertex never point to another red vertex. This means that the possible vertex arrangements for any subtree rooted with a black vertex (bbb, bbr, rbb, rbr -- for [left child][root][right child]) correspond to 234 trees.
Searching a red-black tree is just the same as searching an ordinary binary tree. Insertion and deletion are similar, except that a "fix-up" rotation may be required at some point to preserve the red-black invariant.
Cheers!
What is the name of the binary tree (or the family of the binary
trees), that is balanced, and has the minimum number of nodes
possible for its height?
balanced binary tree
(data structure)
Definition: A binary tree where no leaf is more than a certain amount farther from the root than any other. After inserting or deleting a node, the tree may rebalanced with "rotations."
Generalization (I am a kind of ...)
binary tree.
Specialization (... is a kind of me.)
AVL tree, red-black tree, B-tree, balanced binary search tree.
Aggregate child (... is a part of or used in me.)
left rotation, right rotation.
See also BB(α) tree, height-balanced tree.
-- http://www.itl.nist.gov/div897/sqg/dads/HTML/balancedbitr.html
It is called Fibonacci tree
AVL is a balanced tree with log(n) height (this is the lowest height possible for binary tree).
Another implementation of a similar data structure is Red Black Tree.
Both trees implement all operations in O(log(n)).
AVL Tree is something that you have been looking for.
From Wikipedia:
In computer science, an AVL tree is a self-balancing binary search tree, and it is the first such data structure to be invented. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; therefore, it is also said to be height-balanced. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.