Wavelet-domain hidden Markov models have proven to be useful tools for statistical signal and image processing. The hidden Markov tree (HMT) model captures the key features of the joint probability density of the wavelet coefficients of real-world data. One potential drawback to the HMT framework is the need for computationally expensive iterative training to fit an HMT model to a given data set (e.g., using the expectation-maximization algorithm). We greatly simplify the HMT model by exploiting the inherent self-similarity of real-world images. The simplified model specifies the HMT parameters with just nine meta-parameters (independent of the size of the image and the number of wavelet scales). We also introduce a Bayesian universal HMT (uHMT) that fixes these nine parameters. The uHMT requires no training of any kind, while extremely simple, we show using a series of image estimation/denoising experiments that these new models retain nearly all of the key image structure modeled by the full HMT. Finally, we propose a fast shift-invariant HMT estimation algorithm that outperforms other wavelet-based estimators in the current literature, both visually and in mean square error.