Permit me to include this nice image from Day & Li
to illustrate @Igor's point that "in general on a surface, it [the cut locus] is a graph not a tree."
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<img src="https://i.sstatic.net/fEQTr.png" width="250" />
<br />
<sup>
The source point $p$ is on the cat's forehead,
the other side in this rear-view.
</sup>
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>Dey, Tamal K., and Kuiyu Li. "Cut locus and topology from surface point data." *In Proceedings of the 25th Symposium on Computational Geometry*, pp. 125-134. 2009.
[ACM link](https://dl.acm.org/doi/abs/10.1145/1542362.1542390).

In the paper they compute an approximation to the cut locus
on this model that pretty much follows the smooth curves
drawn above.

  [1]: https://i.sstatic.net/fEQTr.png