Polymers often consist of long non-selfintersecting but fluctuating paths of monomers connected by bonds and immersed in a solvent. In the simplest models path probabilities are (normalized) products of independent positive random weights (exponentials of random potentials) reflecting impurities (heterogeneities) in the solvent. Of interest are the statistical laws describing the polymer path fluctuations that can be obtained with certainty. For example, is the end of a long polymer chain SURE to behave according to some identifiable statistical law ? If so, is it universal or, if not, how does it depend on the parameters of the potential (random weights) ? The answer is ``yes, both'', but depending on certain types of disorder (weak or strong). In this talk some recent results, speculation and conjectures will be presented for a simple mathematical model in which the polymer paths are represented by random paths of vertices in a rooted, directed binary tree. The aim is partially to highlight some well-defined problems for computational/statistical exploration and analysis. Much of this is based on joint work with Stanley Williams and Torrey Johnson.