We present a graph theoretical approach to the configurational statistics of random treelike objects, such as randomly branching polymers. In particular, for ideal trees we show that Prüfer labeling provides (i) direct access to the exact configurational entropy as a function of the tree composition, (ii) computable exact expressions for partition functions and important experimental observables for tree ensembles with controlled branching activity, and (iii) an efficient sampling scheme for corresponding tree configurations and arbitrary static properties.
