$(20$ pts$)$ A rooted binary tree has three types of nodes, red, green, and blue with the following

properties:

Each red node has two children, exactly one of which is green.

Each green node has exactly one child, which is not green.

Blue nodes have no children.

Let $R,G,$ and $B$ respectively denote the number of red, green, and blue nodes. Prove $($by

induction on the tree height$)$ that

$($a$)B=R+1$

$($b$)$ either $G=R$ or $G=B$; and

$($c$)G=B$ if and only if the root is green.