The correct expression for the total number of binary trees possible with height n - 2 and n nodes is:
(2n - 3)^(n - 2)
Explanation:
In a binary tree, each node can have at most two child nodes - a left child and a right child. For a binary tree with n nodes, the height of the tree is the maximum number of edges from the root to a leaf node.
In this case, we are given that the height of the binary tree is n - 2. So, the height of the tree is less than the total number of nodes, ensuring that the tree is not a degenerate tree (where one branch dominates the other).
The number of possible binary trees with n nodes and a given height is related to the Catalan numbers. The Catalan number C(n) gives the number of structurally unique binary trees with n nodes.
The expression (2n - 3)^(n - 2) represents the total number of binary trees possible with height n - 2 and n nodes. It can be derived by considering that for each internal node (excluding the leaf nodes), there are two choices for the left and right child nodes, resulting in (2n - 3) choices. Since there are (n - 2) internal nodes in a tree with n nodes and height n - 2, we raise (2n - 3) to the power of (n - 2).
Therefore, the correct expression is (2n - 3)^(n - 2). None of the provided options match this expression.
