I tried finding the difference between m way tree and the m way search tree. Most resources only tells about m way search tree and end up being on B tree or B+ trees.
My doubts are:-
Is it analogous to the binary tree and binary search tree?
I read somewhere that m way trees don't have any particular order and
every node has to be filled fully before moving to the new node.(complete
tree)
Is it analogous to the binary tree and binary search tree?
Yes
m way trees don't have any particular order
This is true
and every node has to be filled fully before moving to the new node.(complete tree)
Something like this describes a step in an algorithm, and has little to do with the data structure itself: nothing is "moving" in a data structure.
Definitions
In short: an m-way tree puts no conditions on the values stored in the nodes, while an m-way search tree does.
Reva Freedman, associate professor at Northern Illinois University has notes on Multiway Trees where four terms are defined in succession, each time indicating which additional requirements apply for the next term:
multi way tree,
m-way tree
m-way search tree
B-tree of order m
Multiway Trees
A multiway tree is a tree that can have more than two children. A
multiway tree of order m (or an m-way tree) is one in which a tree can
have m children.
As with the other trees that have been studied, the nodes in an m-way
tree will be made up of key fields, in this case m-1 key fields, and
pointers to children.
To make the processing of m-way trees easier, some type of order will
be imposed on the keys within each node, resulting in a multiway
search tree of order m ( or an m-way search tree). By definition an
m-way search tree is a m-way tree in which:
Each node has m children and m-1 key fields
The keys in each node are in ascending order.
The keys in the first i children are smaller than the ith key
The keys in the last m-i children are larger than the ith key
M-way search trees give the same advantages to m-way trees that binary
search trees gave to binary trees - they provide fast information
retrieval and update. However, they also have the same problems that
binary search trees had - they can become unbalanced, which means that
the construction of the tree becomes of vital importance.
B-Trees
An extension of a multiway search tree of order m is a B-tree of
order m. This type of tree will be used when the data to be
accessed/stored is located on secondary storage devices because they
allow for large amounts of data to be stored in a node.
A B-tree of order m is a multiway search tree in which:
The root has at least two subtrees unless it is the only node in the tree.
Each nonroot and each nonleaf node have at most m nonempty children and at least m/2 nonempty children.
The number of keys in each nonroot and each nonleaf node is one less than the number of its nonempty children.
All leaves are on the same level.
Related
Just been learning about binary trees in school, and two rules of binary trees is that
every node has at most 2 child nodes
there exists linear ordering defined for the children of each node (ordered pair)
Now, all types of binary trees (full, complete, etc.) are binary trees so they must satisfy these 2 conditions.
However, I saw on GeeksForGeeks this example:
How is 'linear ordering', ordered pair, defined here?
It seems that for the sibling nodes in this picture, some left ones are larger than the right one, some right nodes are larger than the left one.
If asked to check if a given tree is a binary tree, how do I make sure of the second property, that the children of each node have to be ordered?
Thanks
This is one of the complicated ways to introduce a binary tree.
two rules of binary trees is that
every node has at most 2 child nodes
there exists linear ordering defined for the children of each node (ordered pair)
Simple ways of introducing binary trees I could think of are "at most two children and no cycles" or "at most two children and unique path between any pair of vertices".
But fine. You bring up the point of linear order. Lets discuss that.
Here
A linear ordering on a finite collection of objects may be described
as follows: each object has exactly one immediate predecessor object
and one immediate successor object with two exceptions: A first object
has no predecessor and a last object has no successor.
If you have learnt about traversal so far, with the above definition, I would take binary tree traversals as linear order - preorder, postorder, inorder, level order. This applies to all types of binary trees (full, complete, etc.) which includes the complete binary tree you posted as an image.
I'm reading about AVL tree and there I redirected to self balancing tree there I read that
In computer science, a self-balancing (or height-balanced) binary
search tree is any node-based binary search tree that automatically
keeps its height (maximal number of levels below the root) small in
the face of arbitrary item insertions and deletions.1
These structures provide efficient implementations for mutable ordered
lists, and can be used for other abstract data structures such as
associative arrays, priority queues and sets.
I'm confused
What is the relation of height to list? array?
how small height tree provide efficient implementations for mutable ordered lists? array? queues?
Let suppose the node of list or index of array are height of list or
array, how it could be small?
Here is a visual on balance
2. Remember that trees offer a unique path to every node. Because AVL trees are binary search trees we "know" which direction to go when given a value, because the tree is sorted. AVL trees (for every node, the heights of the left and right subtrees differ by at most 1) allow for quick searches and insertions, usually log N, because it can rotate and manipulate itself in constant time. We can implement an ordered list, queue, etc. values into a tree structure to leverage its characteristics. The key is the tree remaining balanced (stays horizontal not vertical, which it does by following the binary search tree trait).
Can you elaborate on 1 and 3?
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
A splay tree is a type of self-adjusting binary search tree. Inserting a node into a splay tree involves inserting it as a leaf in the binary search tree, then bringing that node up to the root via a "splay" operation.
Let us say a binary search tree is "splay-constructible" if the tree can be produced by inserting its elements into an initially empty splay tree in some order.
Not all binary search trees are splay-constructible. For example, the following is a minimal non-splay-constructible binary search tree:
What is an efficient algorithm that, given a binary search tree, determines whether it is splay-constructible?
This question was inspired by a related question regarding concordance between AVL and splay trees.
More details: I have code to generate a splay tree from a given sequence, so I could perform a brute-force test in O(n! log(n)) time or so, but I suspect polynomial time performance is possible using some form of dynamic programming over the tree structure. Presumably such an algorithm would exploit the fact that every splay-constructible tree of size n can be produced by inserting the current root into some splay-constructible tree of size n-1, then do something to take advantage of overlapping/isomorphic subproblems.
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.