We define the notion of a weakly pointed tree, and characterize the amount of genericity necessary to prevent a uniformly branching tree being weakly pointed. We use these ideas to show there is no topological analogue of a measure-theoretic selection theorem of Graf and Mauldin.
"Weakly pointed trees and partial injections." J. Symbolic Logic 73 (1) 363 - 368, March 2008. https://doi.org/10.2178/jsl/1208358757