Final answer:
The maximum height of a 2-3 tree with 16 keys can be determined by the properties of the tree's structure. A 3-level 2-3 tree, using 3-nodes, can accommodate up to 18 keys, therefore, with 16 keys, the tree's maximum height would be 3.
Step-by-step explanation:
To determine the maximum height of a 2-3 tree with 16 keys, we must understand the properties of a 2-3 tree. Firstly, a 2-3 tree is a balanced search tree with nodes that can have either two children (2-node) or three children (3-node). In either case, a node will hold one key (for a 2-node) or two keys (for a 3-node). Since it's a balanced tree, it has the property that all leaf nodes are at the same height, which ensures the tree remains as flat as possible as it grows wider instead of taller.
With the understanding of the 2-3 tree structure, we can calculate the maximum height with 16 keys. Considering the best-case scenario, where the tree is minimally tall and maximally wide, each node would be a 3-node (holding the maximum number of keys), and all leaf nodes would be at the same level of the tree. The first level (root) would hold 2 keys, the next level could accommodate 2 * 3 = 6 keys (since each parent could lead to 3 children), and thus, in the third level, we could accommodate a total of 6 * 3 = 18 keys. This allows all 16 keys to fit within three levels, hence the maximum height of the tree could be 3.