What are some examples of how Stockton uses euphemisms to tell readers that the king is semi-barbaric in "The Lady or the Tiger?"

The word "semi-barbaric" means uncivilized; therefore, the king in "The Lady or the Tiger?" is prone to resort to uncivilized methods of justice. One description of the king is as follows:

". . . for nothing pleased him so much as to make the crooked straight, and crush down uneven places."

Since a euphemism is a nice, mild, or polite word or phrase substituted for a more common, blunt, or offensive one, the above passage...

The word "semi-barbaric" means uncivilized; therefore, the king in "The Lady or the Tiger?" is prone to resort to uncivilized methods of justice. One description of the king is as follows:

". . . for nothing pleased him so much as to make the crooked straight, and crush down uneven places."

Since a euphemism is a nice, mild, or polite word or phrase substituted for a more common, blunt, or offensive one, the above passage is politely saying that the king is a control-freak who wants everything to go his way.

Another turn of phrase that makes the king seem nicer than reality is the following:

"The king allowed no such subordinate arrangements to interfere with his great scheme of retribution and reward."

This means that no other negotiations could take place for the "criminal" who must face the arena; and, the punishment is never equal to the crime, but either life or death. The king is swift to send someone to the arena and satisfied for the person to be pitted against chance in order to pay for his crime.

Finally, one last example of euphemism is the description of the king when he discovers that his daughter has been dating a man of low class:

"He did not hesitate nor waver in regard to his duty in the premises. The youth was immediately cast into prison, and a day was appointed for his trial in the king's arena."

The king is not a patient man. He doesn't negotiate, as mentioned above, but he also does not listen to those beneath him. Without patience, a person would react in a barbaric way. It's the use of reason, communication, and mercy that sets rational humans apart from the barbaric ones.