**Exercise:**A tree has 2n vertices of degree 1. 3n vertices of degree 2 and n vertices of degree 3. Determine the number of vertices and edges in the tree.

**Solution:**

2n vertices has total degree = 2n× 1 = 2n

3n ″ ″ ″ ″ = 3n×2 =6n

__n ____″ ____ ____″ ____ ____″ ____ ____″ = n×3 =3n __

6n ″ ″ ″ ″ =11n [ By adding]

Since the sum of the degree of the vertices of a tree with n vertices is

2(n-1) = 2n-2

∴ 11n = 2.6n-2

⟹11n=12n-2

⟹12n-11n=2

⟹ n=2

Therefore the total number of vertices is v= 6n = 6×2=12 (Answer)

And the total number of edges is e = v-1 = 12-1 = 11 (Answer)

## Exercise:A tree has two vertices of degree 2. One vertex of degree 3 and 3 vertices of degree 4. How many vertices of degree 1 does it have?

**Solution:**

2 vertices has total degree = 2× 2 = 4

1 ″ ″ ″ ″ = 1×3 =3

3 ″ ″ ″ ″ = 3×4 =12

__n ____″ ____ ____″ ____ ____″ ____ ____″ = n×1 =n __

6+n ″ ″ ″ ″ =19+n [ By adding]

Since the sum of the degree of the vertices of a tree with n vertices is

2(n-1) = 2n-2

∴19+n= 2. (6+n) -2

⟹19+n=12+2n-2

⟹12+2n-2=19+n

⟹ 2n-n=19+2-12

⟹n=9

Hence the number of vertex of degree 1 is 9. **(Answer)**