An important size bias change of measure was introduced in the context of multiplicative cascades by Jacques Peyriere, also referred to as the 'Peyriere probability', for the purpose of analyzing the structure of random measures obtained in the supercritical regime in which non-trivial cascade limits can be determined from the seminal 1976 paper of Kahane and Peyriere. Size biasing was extended by Stanley Williams and the author in 1994 as another approach to the Kahane-Peyriere existence theory for multiplicative cascades. Certain problems arising in the analysis of tree polymers will be shown to involve the analysis of cascade limits in the critical and subcritical cases, to which size biasing can effectively be applied. Some improvements on present theory are possible by this approach which will be presented. Based on joint work with Stanley C Williams, and inspired by discussions with Harry Kesten during a sabbatical year at Cornell.