Since 1513, Denmark has alternated kings called Frederick and Christian. What is the reason for this and how has no king in over 500 years broken the tradition?

The simple answer is tradition. The royal house of Denmark has kept a non-violent transition of power since Frederik I ousted his nephew, Christian the second, in 1523. Christian the second's father was king Hans, and the last male danish monarch who didn't bear the name Christian og Frederik.

During the rennaisance, it became the norm for European Monarchs to keep to a few names in the line of kings in a given country, to the degree were new kings would change their name if they succeeded to the throne without the proper name. A late example of this is king george VI of England. George, who was initially named Albert, wasn't expected to succeed to the throne, being the younger brother, but, upon his brother's resignation from the Throne, he took the name George, to keep the tradition of the royal names (George, Edward, William and James). You see the same in france (Philip, Louis) and most other European monarchies. This tradition originated in the middle ages, where it was common for noblemen, and kings especially, to be named after one of their forefathers, usually another king. This is evident in the danish line of kings, where you initially only find Gorm, Harald, Svend and Knud (Svein and Canute in English, because of inferior pronounciation). These names are used exclusively, until the sons of Svend Estridsen come into power. Svend had more than 20 children, and at least 10 of these where males. Of these 10 males, 5 (including the youngest of them) became kings. At this time, name changes weren't normal, and it would have been ridiculous to have 10 boys with only 4 names to go around. So besides an obligatory Harald, Knud, and two Svends, there was an Oluf and a Niels, who both became kings. What folowed was a series of civil wars, which lasted throughout the 11th and 12th centuries. The winners of these civil wars usually wouldn't name their sons after their rivals, which led to the abandonment of Svend and Harald, and to the Ascension of Valdemar (who was named after his slavic maternal grandfather king Vladimir). 3 more Valdemars followed, and it is still a rather normal middle name for the danish princes. Now, during the middle ages, the kings were usualy warrior-kings, meaning they fought, and often died, in battles. Between 1067 and 1387, literally half of the danish kings died violently. But during the rennaisance, the kings became court-kings instead. This, along with the increasing reliance on using other generals than the king, and keeping him away from the front lines, meant that no danish king in the last 500 years died violently. Not even Christian the 2nd, who was ousted,but allowed to live out his life in imprisonement. This resulted in a much more stable line of succesion, which coincidentally resulted in a much more stable and limited list of names.

Sauce: for the medieval kings, Gesta Danorum, by Saxo Gramatticus, along with almost anything by Niels lund, Nils Hybel and Michael Gelting. For the rennaisance kings i can recommend almost any book by Sebastian Olden-Jørgensen.

Hope this answered your question, and doesn't get.... modded. :)