Just for Godfried: a forest of random Pythagoras trees. Here is my Postscript code for one of the trees.
I took each random vertex uniformly on a halfcircle, but one could consider other distributions as well. Here are some theoretical questions related to it.

 


 


 


 


 


 


 


 


 
In the last three trees, I flipped triangles to make the largest branch always grow most vertically up. 
Contact 
Luc Devroye 