In this paper, we mainly study a class of small deviation theorems for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment. Firstly, we give the definition of Markov chains indexed by a tree with uniformly bounded degree in random environment. Then, we introduce the some lemmas which are the basis of the results. Finally, a class of small deviation theorems for functionals of random fields on a tree with uniformly bounded degree in Markovian environment is established.
