We present a deterministic operator on tree codes - we call tree code product - that allows one to deterministically combine two tree codes into a larger tree code. Moreover, if the original tree codes are efficiently encodable and decodable, then so is their product. This allows us to give the first deterministic subexponential-time construction of explicit tree codes: we are able to construct a tree code T of size n in time 2 nε. Moreover, T is also encodable and decodable in time 2 nε. We then apply our new construction to obtain a deterministic constant-rate error-correcting scheme for interactive computation over a noisy channel with random errors. If the length of the interactive computation is n, the amount of computation required is deterministically bounded by n 1+o(1), and the probability of failure is n -ω(1).