  The hierarchical and recursive expressive capability of rooted trees is
applicable to represent statistical models in various areas, such as data
compression, image processing, and machine learning. On the other hand, such
hierarchical expressive capability causes a problem in tree selection to avoid
overfitting. One unified approach to solve this is a Bayesian approach, on
which the rooted tree is regarded as a random variable and a direct loss
function can be assumed on the selected model or the predicted value for a new
data point. However, all the previous studies on this approach are based on the
probability distribution on full trees, to the best of our knowledge. In this
paper, we propose a generalized probability distribution for any rooted trees
in which only the maximum number of child nodes and the maximum depth are
fixed. Furthermore, we derive recursive methods to evaluate the characteristics
of the probability distribution without any approximations.
