Theoretical Proof on Red$-$Black Trees

Prove by induction that a non$-$empty red$-$black tree with more than one node must contain at least one

red node. Consider the properties and insertion rules of red$-$black trees in your proof, especially focusing

on how the tree maintains its balancing and coloring constraints after insertions.

Your proof should include:

A clear base case.

An inductive hypothesis.

An inductive step that covers all relevant scenarios.