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.
Related
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'm learning to use and Binary trees, and I want to understand it better.
Do we consider a root and a left node to that root to be a tree and consider it unbalanced? that will make the answer to be the number of minimal leaves in a tree to be 1.
Otherwise if a tree needs 2 nodes under the root that will change the answer to the minimal leaves to be 2.
The minimal number of leaves in a binary tree is zero
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 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.
I need to write function, which receives some key x and split 2-3 tree into 2 2-3 trees. In first tree there are all nodes which are bigger than x, and in second which are less. I need to make it with complexity O(logn). thanks in advance for any idea.
edited
I thought about finding key x in the tree. And after split its two sub-trees(bigger or lesser if they exist) into 2 trees, and after begin to go up and every time to check sub-trees which I've not checked yet and to join to one of the trees. My problem is that all leaves must be at the same level.
If you move from the root to your key and split each node so one points at the nodes larger than the key and the other at the rest and then make the larger node be a part of your larger tree, say by having the leftmost node at one level higher point at it, (don't fix the tree yet, do it at the end) until you reach the key you will get your trees. Then you just need to fix both trees on the path you used (note that the same path exists on both trees).
Assuming you have covered 2-3-4 trees in the lecture already, here is a hint: see whether you can apply the same insertion algorithm for 2-3 trees also. In particular, make insertions always start in the leaf, and then restructure the tree appropriately. When done, determine the complexity of the algorithm you got.