So as the title says I want to prove the above using induction.
Let T be a nonempty, full binary tree then we have a full binary tree with $L=I+1$. We will show that the above will still be valid if we add $2$ nodes pointing to $2$ new vertices in every existing leaf.
By doing the above change:
i) we remove a leaf, as to become an internal node.
ii) we add a new internal node.
iii) we add two new leafs.
• Number of leafs = (number of old leafs) $-1 +2$
• Number of internal vertices= (number of old internal nodes) $+1$
The above will apply every time for all the new leafs of the old tree. If we do that for n leafs, it will increase by n the number of 2 leafs over the number of the nodes.
So now we have $L+n =I+n+1 \to L = I+1$. Proved.
Is my process of thought correct?