While there has been a growing body of work in child-robot interaction, we still have very little knowledge regarding young children's speaking and listening dynamics and how a robot companion should decode these behaviors and encode its own in a way children can understand. In developing a backchannel prediction model based on observed nonverbal behaviors of 4-6 year-old children, we investigate the effects of an attentive listening robot on a child's storytelling. We provide an extensive analysis of young children's nonverbal behavior with respect to how they encode and decode listener responses and speaker cues. Through a collected video corpus of peer-to-peer storytelling interactions, we identify attention-related listener behaviors as well as speaker cues that prompt opportunities for listener backchannels. Based on our findings, we developed a backchannel opportunity prediction (BOP) model that detects four main speaker cue events based on prosodic features in a child's speech. This rule-based model is capable of accurately predicting backchanneling opportunities in our corpora. We further evaluate this model in a human-subjects experiment where children told stories to an audience of two robots, each with a different backchanneling strategy. We find that our BOP model produces contingent backchannel responses that conveys an increased perception of an attentive listener, and children prefer telling stories to the BOP model robot.