Question

I need Help Plz

In a tree, the leaves are called external nodes. Accordingly, internal nodes are exactly the nodes that are not external nodes. An edge or connection exists between two nodes if the two nodes are in `` father-child relationship ''.
A true binary tree is a tree with the property that every internal node has exactly two children.

Prove the following two sentences for nonempty real binary trees:
a) A non-empty real binary tree with N internal node has N + 1 external nodes. (Hint: Take advantage of the fact that the induction requirement applies in particular to true subtrees and (induced) subtrees of binary trees are themselves binary again.)
b) A real binary tree with N internal node has 2N edges.

Answer 1

a) Consider an example for the binary tree containing nodes namely - 8,5,4,6,3.

In the above given binary tree, nodes 4 and 5 are the internal nodes, which contains the external nodes 3,6 and 8 respectively.So it satisfies the condition N which contains N + 1 such that total number of internal nodes is n = 2 contains external nodes which is (n + 1) = 2 + 1 = 3.

b) A full binary tree contains exactly less than or equal to 2 child nodes.Thus, it concludes a full binary tree with internal nodes has 2n edges.Since a tree has a one more vertex than it has edges.A full binary tree contains 2n + 1 vertices, n + 1 external nodes and 2n edges.This can also be computed by using the given formulas.

Consider an example with n = 1, which contains 2 edges, 3 vertices and 2 external nodes

   

By n = 2 can be calculated by using the formula, has 4 edges, 5 vertices and 3 external nodes.Similarly it can be derived for internal node n = 3,4 and so on.

The given example is for internal node n = 2.