Definition of a Balanced Tree

Question

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?

Answer 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.

Answer 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)