UES elementary schools - how does this work?

I'm considering a move to NYC with my 6 y/o daughter (will be in first grade in fall of 2016). I am a former New Yorker and spent a few years on the UES so I am familiar with many facets of life in NYC but not public school zoning. We will be renting to start. If I rent a place in a certain zone, does that guarantee my daughter a place in the zoned school? It seems I read conflicting accounts of this. If there is overflow at the zoned school, how would her school be determined? Next closest? Etc. Understandably I'm nervous about getting settled this summer and then finding out she can't get into the school in the neighborhood where we rented just to get into the school. Does that make sense? Can anyone help?