A closed form expression for # of inputs of a binary tree and its number of nodes
There is a known expression linking the number of input pins and the number of nodes in a binary tree. It does not use the height of the tree.

For any binary tree the number of leaves equals the number of internal nodes plus one.

We can prove this by induction. Basis. A tree with one node has two leaves. Induction step. Whenever I replace a leaf by a new node, both the number of nodes and the number of leaves increase by one (we lose a leaf, and get two new leaves in return).