We need to arrange the given words from the broadest category to the most specific category.
The broadest term is:
\[
\text{Animal}.
\]
All the given living organisms come under animals.
Among animals, vertebrates are animals having backbone.
So:
\[
\text{Vertebrates}
\]
comes after animals.
Among vertebrates, mammals are a class of animals.
So:
\[
\text{Mammals}
\]
comes after vertebrates.
Among mammals, feline refers to the cat family.
So:
\[
\text{Feline}
\]
comes after mammals.
Finally, cat is a specific animal belonging to the feline family.
So the logical order is:
\[
\text{Animal} \rightarrow \text{Vertebrates} \rightarrow \text{Mammals} \rightarrow \text{Feline} \rightarrow \text{Cat}.
\]
Now match this with the given letters:
\[
D,\ B,\ A,\ E,\ C.
\]
Therefore, the correct answer is:
\[
D,\ B,\ A,\ E,\ C.
\]