There is no confusion/conflict between definition of Perfect Binary Tree,(Full/Strict/Proper) Binary Tree and Balanced Binary Tree.
Source 1 (mycodeschool, GeeksForGeeks):
Complete Binary Tree: A Binary Tree is a complete Binary Tree if all the levels are completely filled except possibly the last level and the last level has all keys as left as possible.
No mention of Almost Complete Binary Tree
Source 2 (My teacher, many YouTubers):
Almost Complete Binary Tree: Same definition of Complete Binary tree from Source 1.
Complete Binary Tree: Same definition as Perfect Binary Tree, that is it's saying Complete Binary Tree and Perfect Binary Tree are same.
Why there exists different definitions in terminologies? and which source should I consider?
Related
I know well about Full Binary Tree and Complete Binary Tree. But unable to make Full binary tree with only 6 nodes.
The answer is No. You can't make a Full binary tree with just 6 nodes. As the definition in the Wikipedia says:
A full binary tree (sometimes referred to as a proper or plane
binary tree) is a tree in which every node has either 0 or 2
children. Another way of defining a full binary tree is a recursive
definition. A full binary tree is either:
A single vertex.
A tree whose root node has two subtrees, both of which are full binary trees.
Another interesting property I noticed is that, the number of nodes required to make a full binary tree will always be odd.
Another way to see that a full binary tree has an odd number of nodes:
Starting with the definition of a full binary tree (Wikipedia):
a tree in which every node has either 0 or 2 children.
This means that the total number of child nodes is even (0+2+2+0+...+2 is always even). There is only one node that is not a child of another, which is the root. So considering that node as well, the total becomes odd.
By consequence there is no full binary tree with 6 nodes.
Elaborating on #vivek_23's answer, this is, unfortunately, not possible. There's a beautiful theorem that says the following:
Theorem: Any full binary tree has 2L - 1 nodes, where L is the number of leaf nodes in the tree.
The intuition behind this theorem is actually pretty simple. Imagine you take a complete binary tree and delete all the internal nodes from it. You now have a forest of L single-node full binary trees, one for each leaf. Now, add the internal nodes back one at a time. Each time you do, you'll be taking two different trees in the forest and combining them into a single tree, which decreases the number of trees in the forest by one. This means that you have to have exactly L - 1 internal nodes, since if you had any fewer you wouldn't be able to join together all the trees in the forest, and if you had any more you'd run out of trees to combine.
The fact that there are 2L - 1 total nodes in a full binary tree means that the number of nodes in a full binary tree is always odd, so you can't create a full binary tree with 6 nodes. However, you can create a full binary tree with any number of odd nodes - can you figure out how to prove that?
Hope this helps!
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 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 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"?
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.