The height of a binary tree is the number of edges in the longest path from the root to a leaf in the tree. The maximum possible height of a full binary tree with 23 nodes ________ is . (answer in integer)

The correct option is 11.

Let us understand the definition of a full binary tree and how we can achieve the maximum possible height for a given number of nodes.

A full binary tree (sometimes called a proper or 2-tree) is a binary tree in which every node has either 0 or 2 children. In other words, no node in a full binary tree has only one child.

We are given that the tree has $N=23$ nodes, and we want to find the maximum possible height $h$ of this tree. The height of a binary tree is defined as the number of edges on the longest path from the root to a leaf node.

To maximize the height of a full binary tree with a fixed number of nodes $N$, we must make the tree as skewed as possible. In a full binary tree, to extend the height by 1 at each step while maintaining the full binary tree property, we must add exactly 2 children to a leaf node at each level.

Let us construct the tallest full binary tree step-by-step:
At level 0 (the root): we have 1 node. To make it a full binary tree, it must have 2 children (at level 1), or be a leaf. Since we want to maximize height, we add 2 children.
Now we have 3 nodes in total, and the height is 1.
To increase the height to 2, one of the nodes at level 1 must have 2 children. The other node at level 1 remains a leaf (0 children).
This adds 2 more nodes, making the total number of nodes 5, and the height becomes 2.
Continuing this process, at each step we select one leaf at the deepest level and give it 2 children. This increases the height by 1 and adds exactly 2 nodes to the tree.

In general, for a full binary tree of maximum height $h$, the relationship between the number of nodes $N$ and the height $h$ is given by the formula:
$N=2h+1$

We can rearrange this formula to solve for the height $h$ in terms of $N$:
$2h=N-1$
$h=\frac{N-1}{2}$

Substituting the given value of $N=23$ nodes into the equation:
$h=\frac{23-1}{2}$
$h=\frac{22}{2}$
$h=11$

Thus, the maximum possible height of a full binary tree with 23 nodes is 11.