Here’s an answer from the internet:
‘The largest brain in the world was found in the head of an insane man, and the psychiatrist’s job is to remedy the psychological problems of people, thereby shrinking their head from border line nuts to a regular or average size’.
I’m not sure that this is right, and will be avoiding wikianswers for the foreseeable future.
A wider internet search gives a consensus that psychiatrists are so-called because of a pejorative comparison between them and tribes whose custom it is to shrink the heads of their slain enemies. I almost wrote ‘primitive tribes’ there, but I have no idea how to shrink a head, and it might be very difficult. This ‘head-shrinking’ explanation sounds plausible.
As luck would have it, I joined Camden Libraries* today, and membership has given me access to the complete online Oxford English Dictionary. With this resource at hand I am able to tell you that the term was first used by American novelist Thomas Pynchon in his book ‘The Crying of Lot’, where he says:
‘It was Dr Hilarius, her shrink or psychotherapist’
This is merely the word’s first recorded use and doesn’t means that Pynchon invented the term. He is famously reclusive, and so it’s unlikely I’ll be able to ask anyway. This website suggests ‘magazine cartoons of the 1950’s and 1960’s were awash in cannibalistic natives, witch doctors and the like, so the imagery of "shrink" is not all that surprising’. Like a lot of expressions it was probably in use before a novelist, like Pynchon, with an ear for vernacular, picked it up.
I’m making less progress on the history of ‘trick cyclist’ and all I’ve managed to find out so far is that it’s a ‘humorous alteration of psychiatrist’ (OED again), which I knew already. If anyone knows anything about this please let me know.