The opening line of Stockton's "The Lady, or the Tiger?" states that the king is semi-barbaric, which means that he can be civilized, but he also isn't beyond using violence, or against sacrificing human life in the name of sport, either. His daughter is described to be just like him, as follows:
"The semi-barbaric king had a daughter as blooming as his most florid fancies, and with a soul as fervent and imperious as his own."
This means that the daughter is exactly like him--civilized, yet able to do cruel and unusual things. The daughter is also viewed by her father as "the apple of his eye, and was loved by him above all humanity." Therefore, the king and his daughter probably had a fairly good relationship until she took a lover without her father's consent, a lover who is not a prince. As a result, the king throws the man into the arena and then probably chose a man for her to marry in the near future.
As previously stated, both the king and the princess are semi-barbaric and capable of inflicting uncivilized consequences upon people with whom they disagree. Therefore, the king's relationship with his daughter can go either of two ways: One, the daughter, being semi-barbaric herself, could be understanding of her father's ways, send her lover to the tiger, and move on with her life. Second, being semi-barbaric herself, she could hire someone to kidnap her father and throw him into the arena in order to inflict poetic justice for doing the same thing to her lover. Either way, the king's relationship with his daughter will not be the same after what he does to her lover. She was not brought up to forgive or forget, so she's more likely to take revenge on her father than let things go.