Recently, the structural properties and limit properties of many complex systems such as tree graphs or tree networks have become hot topics in research, especially in the field of tree-index Markov chains. A large number of domestic and foreign scholars have obtained numerous significant research results. The non-homogeneous bifurcating Markov chain indexed by a binary tree is a special kind of tree-index Markov chain, and its limit properties have been extensively studied by scholars and applied to many fields such as biodynamics and information theory. This paper is devoted to the limit theorem of the transition probability for non-homogeneous bifurcating Markov chains indexed by a binary tree and the relationship between this theorem and the tree-index Markov chain model. First of all, under new conditions, we present a strong limit theorem for non-homogeneous branch Markov chains indexed by a binary tree taking values in a finite state space. Moreover, the strong limit theorem for the harmonic average of its random transition probabilities has been obtained. Finally, by using the equivalent of the above two models and the mean inequality, we extend the limit theorem of the random transition probability for the non-homogeneous Markov chain indexed by a tree.