Question

I am working through Algorithms in C++ by Robert Sedgewick and came across the following statement:

The height of a binary tree with N internal nodes is at least lg N
and at most N-1. The best case occurs in a balanced tree with 2^i
internal nodes at every level except possibly the bottom level. If the
height is “h” then we must have

         2^(h-1) < N+1 <= 2^h

since there are N+1 external nodes.

There wasn’t much explanation surrounding the inequality, so my question is: how did the author deduce the inequality and what is it showing exactly?

Thanks!

Answer 1

The inequality 2^(h-1) < N + 1 <= 2^h demonstrates that, for a given height h, there is a range of node quantities that will have h as a minimum height in common. This is indicative of the property: all binary trees containing N nodes will have a height of at least log(N) rounded up to the next integer.

For example, a tree with either 4, 5, 6 or 7 nodes can have at best a minimum height of 3. One less than this range, and you can have a tree of height 2; one more and the best you can do is a height of 4.
If we map out the minimum height for a tree that grows from 3 nodes to 8 nodes using the base 2 logarithms for N and round up, the inequality becomes clear:

log(3) = 1.58 -> 2  [lower bound]

log(4) = 2    -> 3  [2^(h-1)]
log(5) = 2.32 -> 3
log(6) = 2.58 -> 3
log(7) = 2.81 -> 3

log(8) = 3    -> 4  [2^h | upper bound]

It might be useful to notice that the range (made up of N+1 different quantities) is directly related to the number of external nodes for a given tree. Take a tree with 3 nodes and having a height of 2:

     *
    / \
   *   *

add one node to this tree,

    *          *          *          *
   / \        / \        / \        / \
  *   *  or  *   *  or  *   *  or  *   *
 /            \            /            \
*              *          *              *

and regardless of where you place it, the height will increase by 1. We can then keep creating leaf nodes without changing the height until the tree contains 7 nodes in total, at which point, any further additions will increase the minimum possible height once more:

    *
   / \
  *   *
 / \ / \
*  * *  *

Originally, N was equal to 3 nodes, which meant N+1 = 4 and we saw that there were 4 quantities that had a common minimum height.

If you need more information, I suggest you look up the properties of complete and balanced binary trees.