When can Newton's divided differences be $0$?

Question

I want to try and find the possible degrees of the interpolating polynomial of the nodes $(x_j,j), \quad j=0,1,\dots,n$ and $x_j \in \mathbb{N}$. I have a feeling the degree could only be $1$ or $n-1$, but I'm stuck on how to prove it. I tried to evaluate the divided differences $f\left[x_{0}, \ldots, x_{n}\right]=\sum_{j=0}^{n} \frac{j}{\prod_{k \in\{0, \ldots, n\} \backslash\{j\}}\left(x_{j}-x_{k}\right)}$ and see when they would equal $0$, but I can't quite find all the cases for which this would be true.