I am studying red-black trees and I am reading the Cormen's "Introduction to Algorithms" book. Now I am trying to create red-black tree with numbers 1-10 by using the pseudo-code described in the book - RB-INSERT-FIXUP(T, z). Here is the screenshot
Everything was fine until I inserted number "6" into the tree. According to pseudo-code I get the following result
As you can see all red-black tree requirements met, but I am confused because I know that red-black tree should be balanced on each step.
I can manually perform "left-rotate" procedure with "2" and "4" and change the colours. In that case I will get the following result, which is balanced appropriately
So my question is:
Is that all right to have unbalanced tree?, or I missed something during insertion nodes?
This is fine. Red-black trees are balanced, but not necessarily perfectly. To be precise, properties of red-black tree guarantee that the longest path to the leaf (implicit, not shown in your picture) is at most twice as long as the shortest. Shortest one has length 2 (2 -> 1 -> leaf), longest one has length 4 (2 -> 4 -> 5 -> 6 -> leaf), so the invariant does hold.
They are not balanced, because they do not satisfy the balanced tree property:
A binary tree is balanced if for each node it holds that the number of inner nodes in the left subtree and the number of inner nodes in the right subtree differ by at most 1.
Some books call it "approximately balanced", because guaranteed the logarithmic time add/delete/search operations. (The balanced trees are the AVL trees.)
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've been learning and working on implementing a red-black tree data structure. I'm following this article on red-black tree deletion examples and looking at example 5 they have:
When I insert the same nodes into my tree, I get the following:
I understand that red black trees are not unique (I think), therefore both of the above trees are valid since they don't violate any of the properties.
In the example article, after deleting node 1, they get the following:
But after deleting node 1 in my code, I get the following:
Since in my case, node 1 is red, I don't call my delete_fix function which takes care of re-arranging the tree and such. The deletion algorithm I was following simply states to call a delete_fix function if the node to be deleted is black.
However, after comparing my tree with the one in the example article I can see that mine is not exactly optimized. It still follows the rules of the red-black tree though. Is this to be expected with red-black trees or am I missing something here?
However, after comparing my tree with the one in the example article I can see that mine is not exactly optimized.
It is optimised. Your tree will be fast at deleting nodes 5, 7, 20 & 28. The other only 5 & 7.
Bear in mind that for Red-Black Trees, they can be bushy in one direction. If the black tree height of real nodes is N, then the minimum path from root to leaf node is N (all black) and maximum path from root to leaf node is 2 * N (alternatively black-red-black-red etc). If you try to add a new node to the bushy path that is at maximum height, the tree will recolour and/or rebalance.
If you want a more balanced search tree you should use an AVL tree. Red-Black trees favour minimal insertion/deletion fixups over finding a node. Your tree is fine.
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.
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
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.