This drives me crazy. Here's another example. In 861, Louis the Stammerer rebels against Charles the Bald (king of West Francia), runs off to join the Bretons in Brittany, is given a band of Bretons, and pillages his way through northern France. He is promptly rebuffed by Count Robert the Strong, who fights the Bretons, scatters them, kills many, and drives Louis to flight. Then within the same year, Louis comes home, asks forgiveness for his "excesses", swears "most strict and solemn oaths" to be good in the future, and is given the county of Meaux and the abbacy of St. Crispin as a don't-do-that-again present. He's also ordered to come in person from Neustria along with his wife. (Source: Annals of St. Bertin.)

Reading that makes you want to scream at the page, why did they do that??? The best answer I can give you is that there often wasn't another option because the kings' political position was so fragile. Can't speak to Otto, but Charles the Bald was perennially sitting on a huge number of restive nobles and abbots who wanted more land/money/power then he wanted to give them. Just three years before LTS went on his jaunt, neighboring king (and stepbrother) Louis the German invaded West Francia with the open support of a large number of West Frankish nobles. Other rebellions, defections, and other malfeasance were really common. For most of his 37-year reign, Charles was constantly touring his realm, visiting counts, eliciting oaths of loyalty, and otherwise trying to make sure that they just stayed put and stayed loyal.

In that context... aside from forgiving him, if your son rebels, what exactly are you going to do about it once you put him down? You can't kill him--there's a taboo against shedding royal blood (Edited via /u/thejukeboxhero; if there was a taboo, I couldn't find evidence of it in my files, so strike that), and anyway that's not a precedent you want to be setting. If you imprison him in a monastery, which happened sometimes to out-of-favor royals (Charles's dad was imprisoned twice when his sons rebelled against him), he's just going to become a rallying point for disgruntled nobles who think he'll give them a better deal then you do, and when they rally around him you'll have the same problem you had before except worse. If you keep him with you at all times, he could still get up to mischief and it's a constant headache for you. And anyway you still need people to rule parts of the kingdom for you, because your kingdom is the size of France and it's the year 861 and there's no communication that's faster than a horse. You can't afford to let your son go to waste; you need someone who's at least slightly more trustworthy then whatever non-family schmuck you'd have to dredge up to run Aquitaine instead of him.

I'm sure it galled Charles to forgive Louis, and Otto to forgive his sons as well as Conrad and Liudulf, but there often wasn't another option. The same thing applies to the dukes on a lesser scale. You have to picture a king in this era as like a guy standing in a yard full of unfriendly dogs. You can handle any one of them if they do something bad, but you have to be careful how you scold them, or the whole pack could turn on you. That's not to say that disobedient nobles (or even kings' sons) couldn't receive extremely harsh punishments, such as being blinded, but the politics of the day are probably why it didn't happen more often.

You should compare this clemency with the Sultans of the east, who would have their sons strangled or beheaded at the first signs of treason, most famously Sultan Suleiman and his son Mustafa which historians argue began the long decline of the Ottoman Empire. Mustafa was a capable administrator and thus good choice for Sultan, however was strangled on Suleimans orders, who would later regret this deeply. Selim II, the eventual successor of Suleiman, was a drunkard and lover of orgies. These are traits seemingly did not abandon the later Ottoman Sultans, who the majority would have all brothers and half-brothers strangled upon succession in order to secure their throne.

I literally just wrote my German history bibliographical essay; Gerhard Ritter (quite the character in German historical scholarship), when writing about Frederick the Great (in "Frederick the Great: A Historical Profile") mentions the fact that Frederick's father spared his son when he tried to run away to England; yet his companion (Kitte? I don't have it right in front of me) was put down before his eyes. Ritter mentioned something along the lines of the fact that his son was destined to be a king and needed to learn how to be a "soldier-king" like he was; Frederick was more interested in music and the arts until the death of his friend.

I'll provide sources when I get home, but I believe the gist of it is that these are their sons after all; some are destined to lead in the near future, so they couldn't exactly put down the heir.

For my further reading, could you give some of the examples from the book? As in which kings etc?