Are these two graphs (trees) isomorphic? 

I would say they are not isomorphic since the node with degree 4 is in two different positions, any suggestion?
 A: As Théophile and Bram28 say, you are indeed correct, but you do need to explain this in a more technical way.
I find that the other two answers are a bit too complicated though...  more so than they need be. To show that they are non-isomorphic, all that you need to do is say that in the second graph, the degree-$4$ vertex is adjacent to three leaves (degree-$1$ vertex) and in the first, it is adjacent to only two leaves.
A: You are correct. To make this a bit more 'mathematically acceptable': By definition of graph isomorphism, for them to be isomorph, there needs to be a bijection $f$ between the nodes of the two graphs such that nodes $i$ and $j$ in Graph 1 are connected if and only if $f(i)$ and $f(j)$ in Graph 2 are connected. 
So, to possibly get such a mapping, you clearly have to map node 4 of Graph 1 to node 1 of Graph 2, but now you have exactly 2 nodes in Graph 1 that are adjacent to node 4 in graph 1, so you also need exactly 2 nodes in Graph 2 that are adjacent to node 1 in Graph 2, but in fact you have 3 nodes like that, so no isomorphism is possible.
A: You're right, but your answer isn't formal enough. (What does it mean to be "in the same position"?) You need to be able to give a clear and concrete explanation in terms of the structure or properties of the graph.
One way to see it: deleting the vertex of degree $4$ splits the first graph into components of size $2,2,1,1$, but splits the second into components of size $3,1,1,1$. 
A: The other answers give examples of proofs that these two tree are not isomorphic. Let me say something more general about proving that two graphs are not isomorphic.
In general, a good way to show that two graphs are not isomorphic is to find some structural property that one graph has but the other doesn't. A structural property is technically a property of a graph that is preserved under isomorphism, but the term can be thought of as any property that doesn't depend on how you draw the graph. In this case, "vertices 2 and 6 are five centimeters away from each other" is not a structural property, but "the vertex of degree four has three adjacent leafs" is a structural property. The best proofs of nonisomorphism find the clearest structural properties.
