Compressive sensing aims to recover a sparse or compressible signal from a small set of projections onto random vectors; conventional solutions involve linear programming or greedy algorithms that can be computationally expensive. Moreover, these recovery techniques are generic and assume no particular structure in the signal aside from sparsity. In this paper, we propose a new algorithm that enables fast recovery of piecewise smooth signals, a large and useful class of signals whose sparse wavelet expansions feature a distinct "connected tree" structure. Our algorithm fuses recent results on iterative reweighted l1-norm minimization with the wavelet Hidden Markov Tree model. The resulting optimization-based solver outperforms the standard compressive recovery algorithms as well as previously proposed wavelet-based recovery algorithms. As a bonus, the algorithm reduces the number of measurements necessary to achieve low-distortion reconstruction.