Adonijah - As with the death of Saul, David's succession resulted in a power-struggle. His eldest living son, Adonijah, had designs on the throne himself.
"Now Adonijah, the son of Haggith exalted himself, saying, 'I will be king.' So he prepared for himself chariots and horsemen with fifty men to run before him." (I Kings 1:5, NASB)
However, before his own death, David chose Solomon (his son by Bathsheba) to be king. Besides David, Solomon had the backing of Zadok the high priest, Nathan the prophet and the majority of the people of Israel (I Kings 1:38-39). This ended any possibility of Adonijah being ruler of Israel. At that point he became concerned for his life.
"And Adonijah was afraid of Solomon, and he arose, went and took hold of the horns of the altar. Now it was told Solomon, saying, 'Behold, Adonijah is afraid of King Solomon, for behold, he has taken hold of the horns of the altar, saying, "Let King Solomon swear to me today that he will not put his servant to death with the sword."' Solomon said, 'If he is a worthy man, not one of his hairs will fall to the ground; but if wickedness is found in him, he will die.'" (I Kings 1:50-52, NASB)
Apparently wickedness was found in him, for Solomon had him executed.
"'Now therefore, as the Lord lives, who has established me and set me on the throne of David my father and who has made me a house as he promised, surely Adonijah shall be put to death today.' So King Solomon sent Benaiah the son of Jehoiada; and he fell upon him so that he died." (I Kings 2:24-25, NASB)