How to predict all non-isomorphic connected simple graphs are there with $n$ vertices 
$(1)\:$How many non-isomorphic connected simple graphs are there with $n$ vertices when n is,
$\qquad(a)\:4\qquad(b)\:5$
$(2)\:$Draw all non-isomorphic, cycle free, connected graphs having six vertices.

For $(1)$ when $n=4$, it's only $6$ case I got. But when it is $5$, I was unable to find out $21$ case. Actually I found $15$ case and here is my question arrive,

Is there any prediction without drawing all of those case$?$ If not then how could someone ensure his/her answer in Exam $(\text{For big enough n})?$

For $(2)$ I have the same situation.
I was thinking there should be other way to predict the answer of this kind question. Any help will be appreciated.
Thanks in advances.

I tried for $(2)$ and got $6$ case$(\text{Sorry for my poor drawing})$, 

I think the main hack will be to find all possible tree.
