Tristan, in another time, had been boundlessly energetic; he’d had the habit of pursuing birds with earnest, exuberant, good intentions. Now, in his later years, he was increasingly deliberate, more reluctant to plunge into thorns, and generally stayed closer to hand. His tendency to range had been quieted.
-- Narrator (chapter 1 paragraph 10)

Importance: Ben’s dogs Tristan and Rex represent two different sides of his personality. This introduction of Tristan, the older dog, represents Ben’s suicidal thoughts and the extent to which he has given up on his life when the novel begins.

… when Rex hurled suddenly over the seat to settle his forepaws again on the rucksack, Ben felt a surge of anger. The dog had to learn his place in things, as Tristan before him had. He had to understand how it was.
-- Narrator (chapter 1 paragraph 53)

Importance: In contrast to Tristan, Rex represents Ben’s desire to live and to continue to contribute with his...