  The recursive and hierarchical structure of full rooted trees is applicable
to represent statistical models in various areas, such as data compression,
image processing, and machine learning. In most of these cases, the full rooted
tree is not a random variable; as such, model selection to avoid overfitting
becomes problematic. A method to solve this problem is to assume a prior
distribution on the full rooted trees. This enables the optimal model selection
based on the Bayes decision theory. For example, by assigning a low prior
probability to a complex model, the maximum a posteriori estimator prevents the
selection of the complex one. Furthermore, we can average all the models
weighted by their posteriors. In this paper, we propose a probability
distribution on a set of full rooted trees. Its parametric representation is
suitable for calculating the properties of our distribution using recursive
functions, such as the mode, expectation, and posterior distribution. Although
such distributions have been proposed in previous studies, they are only
applicable to specific applications. Therefore, we extract their mathematically
essential components and derive new generalized methods to calculate the
expectation, posterior distribution, etc.
