Design a heap data structure that supports adding elements and removing the top element, with the following constraints: (1) The topmost element must always be less than the topmost element in the left and right child subtrees; (2) The left child subtree must have as many or one more element than the right child subtree. The heap should also support a 'min' function (returns the value at the top of the tree) and an 'empty' function (returns an empty tree). Provide a description and implementation of such a data structure.