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By family tree I mean the graph of humans which have ever lived, where there is an edge between parent and child.

Is it possible for a family tree to contain a cycle? I'm thinking not, but I don't know how to prove this. Are there any other interesting graphical properties about family trees?

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    As a directed graph, it seems unlikely without a time machine, but as an undirected graph, you can certain have cousins marrying. – 2017-02-27
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    There is a supposition that all humans have evolved from a common ancestor, called Eve. If that's true, everyone is part of a cycle: take the two paths from your parents to Eve. Along with your edges to your parents, that makes a cycle. – 2017-02-28

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If you define an edge between a married couple, definitely you can have a cycle. For example, if a father marry his daughter.

If you have a tree of a species with one parent per child (asexual reproduction), than by definition you can't have a cycle because any directed edge between vertices (A-->B) means that A existed before B.

Following this causality leads to the impossibility of a cycle.