Definition of a Balanced Tree

I am just wondering if someone might be able to clarify the definition of a balanced tree for me. I have that "a tree is balanced if each sub-tree is balanced and the height of the two sub-trees differ by at most one.

I apologize if this is a dumb question, but does this definition apply to every node all the way down to the leaves of a tree or only to the left and right sub-trees immediately off the root? I guess another way to frame this would be, is it possible for internal nodes of a tree to be unbalanced and the whole tree to remain balanced?

Solution 1:

The constraint is generally applied recursively to every subtree. That is, the tree is only balanced if:

1. The left and right subtrees' heights differ by at most one, AND
2. The left subtree is balanced, AND
3. The right subtree is balanced

According to this, the next tree is balanced:

     A
   /   \
  B     C  
 /     / \  
D     E   F  
     /  
    G  

The next one is not balanced because the subtrees of C differ by 2 in their height:

     A
   /   \
  B     C   <-- difference = 2
 /     /
D     E  
     /  
    G  

That said, the specific constraint of the first point depends on the type of tree. The one listed above is the typical for AVL trees.

Red-black trees, for instance, impose a softer constraint.

Solution 2:

There are several ways to define "Balanced". The main goal is to keep the depths of all nodes to be O(log(n)).

It appears to me that the balance condition you were talking about is for AVL tree.
Here is the formal definition of AVL tree's balance condition:

For any node in AVL, the height of its left subtree differs by at most 1 from the height of its right subtree.

Next question, what is "height"?

The "height" of a node in a binary tree is the length of the longest path from that node to a leaf.

There is one weird but common case:

People define the height of an empty tree to be (-1).

For example, root's left child is null:

              A  (Height = 2)
           /     \
(height =-1)       B (Height = 1) <-- Unbalanced because 1-(-1)=2 >1
                    \
                     C (Height = 0)

Two more examples to determine:

Yes, A Balanced Tree Example:

        A (h=3)
     /     \
 B(h=1)     C (h=2)        
/          /   \
D (h=0)  E(h=0)  F (h=1)
               /
              G (h=0)

No, Not A Balanced Tree Example:

        A (h=3)
     /     \
 B(h=0)     C (h=2)        <-- Unbalanced: 2-0 =2 > 1
           /   \
        E(h=1)  F (h=0)
        /     \
      H (h=0)   G (h=0)      

Solution 3:

There's no difference between these two things. Think about it.

Let's take a simpler definition, "A positive number is even if it is zero or that number minus two is even." Does this say 8 is even if 6 is even? Or does this say 8 is even if 6, 4, 2, and 0 are even?

There's no difference. If it says 8 is even if 6 is even, it also says 6 is even if 4 is even. And thus it also says 4 is even if 2 is even. And thus it says 2 is even if 0 is even. So if it says 8 is even if 6 is even, it (indirectly) says 8 is even if 6, 4, 2, and 0 are even.

It's the same thing here. Any indirect sub-tree can be found by a chain of direct sub-trees. So even if it only applies directly to direct sub-trees, it still applies indirectly to all sub-trees (and thus all nodes).

Solution 4:

Balanced tree is a tree whose height is of order of log(number of elements in the tree).

height = O(log(n))
O, as in asymptotic notation i.e. height should have same or lower asymptotic
growth rate than log(n)
n: number of elements in the tree

The definition given "a tree is balanced of each sub-tree is balanced and the height of the two sub-trees differ by at most one" is followed by AVL trees.

Since, AVL trees are balanced but not all balanced trees are AVL trees, balanced trees don't hold this definition and internal nodes can be unbalanced in them. However, AVL trees require all internal nodes to be balanced.