How many birds did Robert "Birdman of Alcatraz" Stroud keep while on Alcatraz?

None. Stroud had bred and studied birds at the Federal Penitentiary in Leavenworth, Kansas. 

Stroud was imprisoned at Alcatraz from 1942 until 1959. 

It was determined that Stroud was abusing his research privileges and sent to Alcatraz.