if we have no level then number of nodes is 0.
To get maximum nodes for each internal node(node which has child) we must add two children nodes.
Method 1
substitute values of levels and check which option matches for corresponding number of nodes this is fast and easy.
checking option (A) 2$^i$-1
substitute i=1 we get 2$^1$-1 = 2 - 1 =1
substitute i=2 we get 2$^2$-1 = 4 - 1 =3
substitute i=3 we get 2$^3$-1 = 8 - 1 =7
so option (A) is correct.
Method 2
If u see above table after increasing every level we double the nodes of previous level and add 1
i.e. if N(i) is the nodes at i$^t$$^h$ level then it can written as
N(i) = 2N(i-1)+1
where N(i-1) is total nodes till level i-1
also N(0)=0 i.e. if there is no level in tree then there is no tree therefore no node
solving this recurrence by back substitution
N(i-1) = 2N(i-2) + 1
N(i-2) = 2N(i-3) + 1
now back substituting
N(i) = 2[2N(n-2)+1]+1
N(i) = 2$^2$N(n-2)+2+1
.
.
.
going to i$^t$$^h$ level in similar way we get
N(i) = 2$^i$N(i-i)+2$^i$$^-$$^1$+2$^i$$^-$$^2$+....+2+1
N(i) = 2$^i$N(0)+2$^i$$^-$$^1$+2$^i$$^-$$^2$+....+2+1
N(i) = 2$^i$*0+2$^i$$^-$$^1$+2$^i$$^-$$^2$+....+2+1
N(i) = 2$^i$$^-$$^1$+2$^i$$^-$$^2$+....+2+1
this is a G.P whose sum is
1(2$^i$$^-$$^1$$^+$$^1$ - 1 )/2-1 formula for GP
=2$^i$-1
so answer is option (A)