Gaussian process priors are often used in Bayesian inverse problems due to their fast computational properties. However, the smoothness of the resulting estimates is not well suited for modelling functions with sharp changes. We propose a new prior that has the same kind of good edge-preserving properties than total variation or Mumford-Shah but correspond to a well-defined infinite dimensional random variable. This is done by introducing a new random variable T that takes values in the space of 'trees', and which is chosen so that the realisations have jumps only on a small set.