We consider hidden Markov models indexed by a binary tree where the hidden state space is a general metric space. We study the maximum likelihood estimator (MLE) of the model parameters based only on the observed variables. In both stationary and non-stationary regimes, we prove strong consistency and asymptotic normality of the MLE under standard assumptions. Those standard assumptions imply uniform exponential memorylessness properties of the initial distribution conditional on the observations. The proofs rely on ergodic theorems for Markov chain indexed by trees with neighborhood-dependent functions.