Final answer:
To determine the structure of the given Binary Search Tree based on a post-order tree walk, we can follow a step-by-step process. The root node is determined as the last key in the post-order sequence. Then, the remaining keys are divided into two groups - smaller and larger than the root. This process is repeated to construct the left and right subtrees, and they are connected to the root to form the tree.
Step-by-step explanation:
A post-order tree walk prints the keys in the order: 1, 3, 2, 5, 7, 6, 8, 10, 9, 12, 11, 4. To determine the structure of the tree, we can start by finding the root node, which is the last key in the post-order sequence. In this case, the root is 4. Then, we can divide the remaining keys into two groups - those smaller than 4 and those larger than 4. The keys 1, 3, 2, and 5 are smaller than 4, and the keys 7, 6, 8, 10, 9, 12, and 11 are larger than 4.
Next, we can construct the left subtree by repeating the process with the smaller keys. The post-order sequence for the left subtree is 1, 3, 2, and 5. The root of the left subtree is 2 since it is the last key in the left subtree sequence. The keys 1 and 3 are smaller than 2, and the key 5 is larger than 2.
Similarly, we can construct the right subtree using the larger keys. The post-order sequence for the right subtree is 7, 6, 8, 10, 9, 12, and 11. The root of the right subtree is 11 since it is the last key in the right subtree sequence. The keys 7, 6, 8, 10, and 9 are smaller than 11, and the key 12 is larger than 11.
Finally, we can connect the root node (4) with its left and right subtrees (2 and 11) to form the complete Binary Search Tree.