While examining the cigarette and ashtray detritus atop the piano, Lt. Tragg tells Buzzie "I thought I smelled tea." "Tea" was 50's -era slang for marijuana.
When Tragg comes to Danny Ross's apartment to arrest Freddie, he checks out the cigarette that Buzzie is smoking and the ashes in the ash tray. This is a not-so-subtle hint that Tragg suspects Buzzie is smoking pot. This is the first reference to illicit drugs in the series.
Buzzie was played by Bobby Troup, who is most famous for writing the song "Route 66" which has been recorded by numerous artists and bands. Troup was a jazz pianist as well. During the early scene when Danny Ross is holding a gun, and Freddie interrupts his seeming contemplation of suicide with a soliloquy, Buzzie is apparently playing slow variations on the classic jazz song, "Stella By Starlight."