I was looking through pictures of the Bronx Zoo photo gallery earlier, and after I saw a picture of a Storm Stork Egg, I asked myself, how dop they keep track of the animals that aren't truly on exhibnit, (The free roaming birds and flying foxes)? In any Indoor Jungle exhibit, this is a problem. Throuh lots of greenery, how do they know the population of birds and stuff? Its a problem if dead birds just stay dead and rot away. that doesn't help other animals health!
There are keepers. These are merely exhibits and rarely are there inaccessible areas. The keepers can account for all of the free range animals or else they go looking for them. It has worked for 40 years But seriously.... the husbandry challenge with free flight exhibits is more about any attempts at breeding. There can also be compatibility issues but then offending individuals are caught up and removed.
In my experience, zoos do not often put real rarities in these exhibits. Partly because of the problem of monitoring them, partly because of the reduced likelihood of successful breeding and partly because of the difficulty of catching a bird to give medical treatment or to move it to another collection etc. In the past I have seen hummingbirds, red birds of paradise and Picathartes in indoor walk-through exhibits. Nowadays I expect to see species such as Java sparrows, Peking robins, fodies, starlings and pigeons which are attractive, not destructive to plants and not predatory on other birds (including eggs and nestlings).
You asked how do keepers know the populations of "free-ranging" birds inside indoor exhibits (I am assuming that you also question it for walkthrough aviaries). The answer is that we count them, every day, often multiple times a day depending on the protocols of that zoo. Also, a dead bird would very quickly be found and wouldn't be left long enough to rot. We walk and clean the exhibits every day. Part of our job as keepers is to not interrupt your immersive experience with watching us clean (though often we still do need to do so during open hours), so a lot of the exhibit cleaning is done either before the zoo opens or after it is closed for the day.
With birds we use Id color bands to ID individuals and we count numbers everyday at places were we generaly meet them for there breakfast feeds or dinner feeds. I know some zoos have smaller off display aviaries hidden in these aviaries where they feed the birds and thus meet them there and count. It is extremely challenging though and I have no idea ( but would love to know ) how Jurong Birdpark does it for their two big aviaries or how Singapore Zoo does it for the fragile forest enclosure.