Consider the set of full binary trees T defined recursively below:

$$ Base case: A single node r is in T

$$ Recursive Rule: If T$1,$ T$2$ are full binary trees with roots r$1,$ r$2$ and r is a node, then adding

edges from r$1$ and r$2$ to r creates a new full binary tree with root r$.$ Prove that the height of the tree will always be less than or equal to $2$v $2,$ where v represents the number of nodes in the tree.

Note: The height of a tree is the number of edges between the root node and it$$s furthest leaf node.