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.
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.
(From: Data Structures Using C By Aaron M. Tenenbaum):
"A complete binary tree of depth d is the strictly binary tree all of whose leaves are at level d."
So, by that meaning the following tree should not be complete binary tree, right?
http://cs-study.blogspot.de/2012/11/complete-binary-tree.html
But, it is according to wikipedia:
In a complete binary tree every level, except possibly the last, is completely filled, and all nodes in the last level are as far left as possible. It can have between 1 and 2h nodes at the last level h.
Please clarify my confusion.
My interpretation:
On level 1..d-1 there are only nodes, and all of them must exist.
At level d, only leaves exists, and they must be filled from left to right
Nodes without children are not considered leaves on level d-1
What is the maximum difference between any two leaves in an AVL tree? If I take an example, my tree becomes unbalanced, if the height difference is more than 2(for any two leaves), but the answer is the difference can be any value. I really don't understand, how this is possible.Can anyone explain with examples?
The difference in levels of any two leaves can be any value! Definition of AVL describes height difference only on two sub-trees from one node.
So you need to fill subtrees with equal height then add new nodes just to create that single node difference. But nobody said that that subtree doesn't contain some subtrees with the exact same definition. Of course tree is selfbalanced but if we'll be that accurate to not touch it's balance then we can create any height difference between some leaves.
Example with leaf 24 on level 3 and leaf 10 on level 6:
According to the explanation in this Wikipedia article, the balancing operations in an AVL tree successfully aim at rearranging the tree such that the height of any two leaves differs no more than one. This is the key property of the data structure which makes the retrieval of nodes efficient (namely logarithmic in the number of nodes of the tree, as a path from the root to a leaf is traversed in the worst case).
I am having trouble with these questions:
A binary tree with N nodes is at least how deep?
How deep is it at most?
Would the maximum depth just be N?
There are two extremes that you need to consider.
Every node has just a left(or right) child, but not right child. In which case your binary search tree is merely a linkedlist in practice.
Every level in your tree is full, maybe except the last level. This type of trees are called complete.
Third type of tree that I know may not be relevant to your question. But it is called full tree and every node is either a leaf or has n number of childs for an n-ary tree.
So to answer your question. Max depth is N. And at least it has log(N) levels, when it is a complete tree.
I have this tree which, for each node, has exactly 10 childnodes (0-9). Each node has some associated data (say, for example, a name and a tag and a color) which, I guess, isn't important for this question. Each of the childnodes has exactly 10 childnodes. A node can be null (which 'ends' the branch') or contain another node.
To visualize what I'm talking about I made this diagram (fear my paintz0r skillz!):
A black box is a null-node. A white box is a node which contains data and childnodes. As you can see, even the root, each node has exactly 10 childnodes. Because of simplicity and to keep the diagram sane I have drawn some nodes very tiny but you can imagine these tiny nodes being the same.
This structure allows me to traverse a path consisting of digits very quickly: a path of 47352 would lead me down the "orange path" to the final destination; 4->7->3->5 where the final 2 cannot be resolved because that last one is a null-node (although colored red) and contains no childnodes.
My question is pretty simple actually: what is this kind of tree called? I have gone through all trees on Wikipedia's Tree (data structure) lemma and the closest I (think I) could get is the Octree and/or K-ary tree. Along those lines of reasoning my tree would be called a Dectree, Decitree, 10-ary tree or 10-way tree or something. But there might be a better name for this. So: anyone?
K-ary tree with K=10
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.
This is something like B-Tree.