Definition of chain of tree

Question

I am trouble understanding the definition of chain of tree at p15.

Here is a rooted tree. The root is "a".
abc is clearly chain. However, I cannot understand whether bc is chain or not.

a
|
bーd
|
c
At first, I thought chain is only "abc","ab","a","abd".

But looking at the proof of Lemma1.5.5(ii), "bc" should be included. Which is correct?

Answer 1

They are giving an example of a chain as the down-set. So a chain is a subset of the graph with comparable elements(a linear ordering), The down-set of a vertex is an example of a chain. So, indeed bc is chain in the graph. They are completely comparable.