I am studying binary tree now and I read that the definition of complete binary tree in CLRS' book "Introduction to algorithms, 3rd edition" is "A complete k-ary tree is a k-ary tree in which all leaves have the same depth and all internal nodes have degree k." (page 1178)
This makes me confused since in wikipedia and many other books this is the definition of so called "perfect binary tree". Can someone please specify which definition is true?
Really appreciate for your answer!
It's the same thing.
According to Wikipedia:
A perfect binary tree is a binary tree in which all interior nodes have two children and all leaves have the same depth or same level. (This is ambiguously also called a complete binary tree.)
https://en.wikipedia.org/wiki/Binary_tree
This definition is for a perfect k-ary tree. For a binary tree, the definition will be: "A perfect binary tree is a binary tree in which all leaves have the same depth and all internal nodes have degree 2."
However, for a complete binary tree, the definition is: all internal nodes have degree 2 but all leaves may not be in same depth.
This link may help you: Difference between "Complete binary tree", "strict binary tree","full binary Tree"?
Related
I came upon two resources and they appear to say the basic definition in two ways.
Source 1 (and one of my professor) says:
All leaves are at the same level and all non-leaf nodes have two child nodes.
Source 2 (and 95% of internet) says:
A full binary tree (sometimes referred to as a proper or plane binary tree) is a tree in which every node in the tree has either 0 or 2 children.
Now following Source 2,
becomes a binary tree but not according to Source 1 as the leaves are not in the same level.
So typically they consider trees like,
as Full Binary Tree.
I may sound stupid but I'm confused what to believe. Any help is appreciated. Thanks in advance.
There are three main concepts: (1) Full binary tree (2) Complete binary tree and (3) Perfect binary tree. As you said, full binary tree is a tree in which all nodes have either degree 2 or 0. However, a complete binary tree is one in which all levels except possibly the last level are filled from left to right. Finally, a perfect binary tree is a full binary tree in which all leaves are at the same depth. For more see the wikipedia page
My intuition for the term complete here is that given a fixed number of nodes, a complete binary tree is made by completing each level from left to right except possibly the last one, as the number of nodes may not always be of the form 2^n - 1.
I think the issue is, what's the purpose of making the definition? Usually, the reason for defining full binary tree in the way that appears in Wikipedia is to be able to introduce and prove the Full Binary Tree Theorem:
The total number of nodes N in a full binary tree with I internal nodes is 2 I + 1.
(There are several equivalent formulations of this theorem in terms of the number of interior nodes, number of leaf nodes, and total number of nodes.) The proof of this theorem does not require that all the leaf nodes be at the same level.
What one of your professors is describing is something I would call a perfect binary tree, or, equivalently, a full, complete binary tree.
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 have some questions on binary trees:
Wikipedia states that a binary tree is complete when "A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible." What does the last "as far left as possible" passage mean?
A well-formed binary tree is said to be "height-balanced" if (1) it is empty, or (2) its left and right children are height-balanced and the height of the left tree is within 1 of the height of the right tree, taken from How to determine if binary tree is balanced?, is this correct or there's "jitter" on the 1-value? I read on the answer I linked that there could be also a difference factor of 4 between the height of the right and the left tree
Do the complete and height-balanced definitions just apply to binary tree or just any other tree?
Following the reference of the definition in wikipedia, I got to
this page. The definition was taken from there but modified:
Definition: A binary tree in which every level, except possibly the deepest, is completely filled. At depth n, the height of the
tree, all nodes must be as far left as possible.
It continues with a note below though,
A complete binary tree has 2k nodes at every depth k < n and between 2n and 2^(n+1) - 1 nodes altogether.
Sometimes, definitions vary according to convenience (be useful for something). That passage might be a variation which, as I understand, requires leaf nodes to fill first the left side of the deepest level (that is, fill from left to right). The definition that I usually found is exactly as described above but without that
passage.
Usually the definition taken for height-balanced tree is the one you
described. In other words:
A tree is balanced if and only if for every node the heights of its two subtrees differ by at most 1.
That definition was taken from here. Again, sometimes definitions are made more flexible to serve specific purposes. For example, the definition of an AVL tree says that
In an AVL tree, the heights of the two child subtrees of any node
differ by at most one
Still, I remember once I had to rewrite an algorithm so that the tree
would be considered height-balanced if the two child subtrees of any
node differed by at most 2. Note that the definition you gave is recursive, this is very common for binary trees.
In a tree whose number of children is variable, you wouldn't be able to say that it is complete (any parent could have the number of children that you want). Still, it can apply to n-ary trees (with a fixed amount of n children).
Do the complete and height-balanced definitions just apply to binary
tree or just any other tree?
Short answer: Yes, it can be extended to any n-ary tree.
Perhaps it is a dumb question but I am confused with the following:
Are n-ary trees and m-way trees the same thing?
I see in various posts talk about m-way trees and other n-ary trees but I am not sure if these are the same data structures or are used for the same problems
Are they the same thing? Could someone please help me clear this out in my head?
From Wikipedia:
In graph theory, a k-ary tree is a rooted tree in which each node has
no more than k children. It is also sometimes known as a k-way tree,
an N-ary tree, or an M-ary tree. A binary tree is the special case
where k=2.
So the answer your question is: Yes, it is the same thing.
If n == m, then there is no difference.
We are dealing with a Most similar neigthbour algorithm here. Part of the algorithm involves searching in order over a tree.
The thing is that until now, we cant make that tree to be binary.
Is there an analog to in order traversal for non binary trees. Particularly, I think there is, just traversing the nodes from left to right (and processing the parent node only once?")
Any thoughts?
update
This tree will have in each node a small graph of n objects. Each node will have n children (1 per each element in the graph), each of which will be another graph. So its "kind of" a b tree, without all the overflow - underflow mechanics. So I guess the most similar in order traversal would be similar to a btree inorder traversal ?
Thanks in advance.
Yes, but you need to define what the order is. Post and Pre order are identical, but inorder takes a definition of how the branches compare with the nodes.
There is no simple analog of the in-order sequence for trees other than binary trees (actually in-order is a way to get sorted elements from a binary search tree).
You can find more detail in "The art of computer programming" by Knuth, vol. 1, page 336.
If breadth-first search can serve your purpose then you can use that.