Empty Forest in Graph Theory

Question

I was reading the Kruskal's Algorithm application and there was a statement about starting with an empty forest. WHat should I understand from the term 'empty forest'? Does it mean an edge that is not connected or should I think of it as a forest with 1-component?

Answer 1

Usually an empty object simply doesn't contain any of whatever its highest-level member happens to be. You can talk about forests in terms of nodes and edges, but the conceptual vantage point of a forest is that it contains trees. If it's empty, it just doesn't contain any trees.

By extension then, it wouldn't contain any nodes or edges since a forest only contains nodes or edges if its corresponding trees contain those same nodes or edges.

Extras: I'm not sure of the author's intention here (whether a forest is defined to be an object containing trees or whether a forest is a graph whose connected components are trees), but in the first interpretation you can run into the interesting state of affairs where a forest has no nodes or edges but is still not empty. How you ask? Simple -- the forest contains any positive number of empty trees.