Question

(a) Give a recursive definition of the set of Rooted Trees.

(b) For a rooted tree T recursive definitions for v(T) and e(T) the number of vertices and edges of T respectively. (c) For a rooted tree T, use structural induction to prove that v(T) − e(T) = 1

Answer 1

a. Base elements :- A null element belongs to set of rooted tree which is an empty tree.

Inductive definition :- If tree T₁ and T₂ belongs to set S of rooted tree then tree T created by adding vertex u as root and left subtree being T₁ and right subtree being T₂ , then T will also belongs to S.

b. If T is null then v(T) = 0 and e(T)=0 else e(T) = e(T₁) + e(T₂) and v(T) = v(T₁) + v(T₂) where T₁ and T₂ are left and right subtrees of T.

c. Base case:- When T contains single node. In this case v(T) = 1 and e(T) = 0, hence v(T) - e(T) = 1 satisfies for base case.

Induction hypothesis :- Let v(T) - e(T) = 1 satisfies for every tree T with v(T) = n and hence e(T) = n-1

Inductive step:- Consider tree T with both left subtree T₁ and right subtree T₂ has less than equal to n vertices.

Then v(T) = v(T₁) + v(T₂) + 1 and e(T) = e(T₁) + e(T₂) + 2

Since T₁ and T₂ satisfies induction hypothesis. So

V(T) - e(T) = v(T₁) - e(T₁) + 1 +v(T₂) - e(T₂) -2 = 1 + 1 +1 -2 = 1

Hence induction hypothesis satisfies for tree T with v(T) >= n. Hence the statement is correct.

Please comment for any clarification.