You might find the following paper useful: [Degree sequences of infinite graphs, by Andreas Blass and Frank Harary](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/135582/jlms0010.pdf).

> The degree sequences of finite graphs, finite connected graphs, finite trees and finite forests have all
been characterized. Our present purpose is to provide such characterizations in the infinite case.