Latent tree graphical models are widely used in computational biology, signal and image processing, and network tomography. Here, we design a new efficient, estimation procedure for latent tree models, including Gaussian and discrete, reversible models, that significantly improves on previous sample requirement bounds. Our techniques are based on a new hidden state estimator that is robust to inaccuracies in estimated parameters. More precisely, we prove that latent tree models can be estimated with high probability in the so-called Kesten-Stigum regime with O(log2n) samples, where n is the number of nodes.