  We study the growth of a time-ordered rooted tree by probabilistic attachment
of new vertices to leaves. We construct a likelihood function of the leaves
based on the connectivity of the tree. We take such connectivity to be induced
by the merging of directed ordered paths from leaves to the root. Combining the
likelihood with an assigned prior distribution leads to a posterior leaf
distribution from which we sample attachment points for new vertices. We
present computational examples of such Bayesian tree growth. Although the
discussion is generic, the initial motivation for the paper is the concept of a
distributed ledger, which may be regarded as a time-ordered random tree that
grows by probabilistic leaf attachment.
