All trees with small or narrow leaves are no good as shelter from heavy rain. You will need some trees with broad leaves, such as coffee trees which also have the tendency for the branches to hang down and with their reasonably broad leaves shed the water quite well, and are quite attractive, and do not grow too high - they are ideal for aviaries, once established.  Privet is popular and good for nesting (finches) but no good for protection from the weather. I also believe that providing nesting places without wet weather protection is a waste of time anyhow, as it nearly always rains when you don't want it to.  

Leave the ends of your aviary free of nest sites. Always put them on the sides of your aviaries or shelters. Winds, even in warm weather, will cause draughts in your aviaries so you will need to guard against these.

You will also need to guard against mice, rats, snakes, black ants, cats, and unfortunately thieves.

In the old days of aviculture the thoughts were on trying to provide for birds what was considered what birds needed in their natural surroundings. 

But most of the time aviculturalists were looking at the situation purely from a human viewpoint. I believe that aviaries for birds must be looked at as a totally artificial environment. 

In that situation, looking at it from the birds' point of view we as aviculturists, should provide the best possible conditions for our birds under that situation.  

We know that most birds like to bathe quite often, but we also should know they do not like to be wet for long periods and so we should provide adequate shelter to which the birds can retire and dry off, especially in wet weather.