Final answer:
The minimum height of a full binary tree with 31 vertices is 4, assuming the tree is completely balanced. The maximum height would be 30, assuming the tree is completely skewed. Therefore, the height of a full binary tree ranges from its minimum to its maximum height based on its structure.
Step-by-step explanation:
The minimum and maximum height of a full binary tree with 31 vertices can be calculated based on the properties of a full binary tree. A full binary tree, also known as a proper or strict binary tree, is one in which every node has either two or no children. To find the minimum height, we consider a perfectly balanced tree where all levels are fully filled. The maximum height occurs when the tree is skewed, meaning one child per node except the last.
The height of a binary tree is the number of edges on the longest path from the root to a leaf. For a full binary tree with 31 vertices, the minimum height would be achieved when the tree is completely balanced. A balanced full binary tree with 31 vertices is a perfect binary tree, which means its height is log2(31+1)-1, which is 4 (since 31 is 2^5-1).
The maximum height would be in the case of a skewed tree where all nodes form a chain, resulting in a maximum height of vertices minus 1, which in this case would be 30 (since there are 31 vertices and we start counting height from 0).