I was watching an episode of big bang theory and someone told his mother "you don't want to see your little bird leave the nest". Why he didn't say leaves?
I might be wrong, but I think that verb "See" uses bare infinitive when it takes an object and to infinitive. So it's like this:
you don't want to see your little bird (to) leave the nest.
But to is not shown, and is not supposed to be shown, because it's bare infinitive.
http://www.englishgrammar.org/bare-infinitive/ Something information about bare infinitive
After verbs like see, feel, hear, .... ( verbs related to senses) you can use either a Verb+ing or a bare infinitive. The latter implies a completed action and the former implies an action which was(is) in progress. compare:
I saw him climbing the wall
This means the observer saw the process of the climbing, but it does not mean the observer saw the whole process.
I saw him climb the wall
This means the observer saw a completed process, from start to end, in which the climber climbed the wall successfully.