A desire for certainty is one of Brinker's most prominent traits. He craves clarity. This is why Brinker insists on investigating the incident on the tree and why he is so eager to enlist in the armed services.
Brinker's personality is of the type that can allow for very little confusion or indeterminancy. Things must be decided, clear and plain. This is, in part, why he has such a dislike for Leper, a confused and conflicted person.
When Gene will not provide a clear and plain explanation for what happened at the tree, Brinker feels he must get an answer. He needs to know whether or not Gene is innocent and whether or not Finny should feel angry and want revenge. Brinker simply cannot handle not knowing.
For the most part the conflicts in this novel take place internally and take the form of emotional and psychological conflict. If Brinker were a different person, he may have debated Gene's guilt and innocence quietly in his own mind. This is not a possibility for Brinker because it is not in his nature to keep internal conflicts to himself. He must externalize problems in order to solve them and he cannot leave a problem unsolved.