Abstract: A uniform spanning tree of 
can be thought of as the ‘‘uniform measure’’ on trees of . The past of 
in the uniform spanning tree is the finite component that is disconnected from infinity when is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length 
, that it has volume at least and that it reaches the boundary of the box of side length around . Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits “mean-field” critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.