Brinker is bothered by the mystery surrounding Finny's fall from the tree where he sustained the injury to his leg. Gene either fails or refuses to provide Brinker with satisfactory details of the accident and Brinker's need to see justice done (along with his desire for order and authority) leads him to organize a trial.
It is clear that Brinker's intent is to conduct an inquiry into the circumstances of Finny's accident.
The trial is held in the Assembly Room and Gene feels that he is being accussed of intentionally causing Finny to fall from the tree.
Though Gene does come to believe that he did, in fact, intentionally shake the limb of the tree to cause Finny to fall, he did not believe this was the case initially. At first, Gene did not know what happened on the tree branch and couldn't say if he had tried to make Finny fall.
At the trial, before Gene is forced to admit to knocking Finny from the tree, Finny rushes out of the Assembly Room and falls down the marble stairs, breaking his leg a second time.