Why did David have the young man put to death?

The chapter gives David's words and no comment of its own. David asks, "Why were you not afraid to stretch out your hand to destroy the LORD’s anointed?" (1:14), and says, "Your blood be on your head, for your mouth has testified against you, saying, ‘I have slain the LORD’s anointed.’" (1:16). They are the words David had used of Saul in 1 Samuel. In the cave he told his men, "The LORD forbid that I should do this thing to my lord, the LORD’s anointed, to stretch out my hand against him, since he is the LORD’s anointed" (1 Samuel 24:6). Over the sleeping Saul he told Abishai, "Don’t destroy him, for who can stretch out his hand against the LORD’s anointed, and be guiltless?" (1 Samuel 26:9). Later David speaks of that day again, to Rechab and Baanah (4:9), who had killed Saul's son Ishbosheth (4:5-8): "when someone told me, ‘Behold, Saul is dead,’ thinking that he brought good news, I seized him and killed him in Ziklag" (4:10). David's words rest on what the young man said himself (1:16). The chapter doesn't say whether his account was true, and it doesn't add a judgment of its own.