A binary tree is full if all of its vertices have either zero or two children. Let $B_n$ denote the number of full binary trees with $n$ vertices.

$($a$)$ By drawing out all full binary trees with $3,5,$ or $7$ vertices, determine the exact values of $B_3,B_5,$ and $B_7.$ Why have we left out even numbers of vertices, like $B_4?$

$($b$)$ For general $n,$ derive a recurrence relation for $B_n.$

$1$

$($c$)$ Show by induction that $B_n$ is $(2^n).$