Ising models and multiresolution quad - trees by

We study percolation and Ising models defined on generalizations of quad-trees used in multiresolution image analysis. These can be viewed as trees for which each mother vertex has 2 daughter vertices, and for which daughter vertices are linked together in d-dimensional Euclidean configurations. Retention probabilities / interaction strengths differ according to whether the relevant bond is between mother and daughter, or between neighbours. Bounds are established which locate phase transitions and show the existence of a coexistence phase for the percolation model. Results are extended to the corresponding Ising model using the Fortuin-Kasteleyn random-cluster representation.