Hi. What did you mean by rows? Are you refering to the vertices that line up together in a certain layout like the top layout?

If so, TreePlot doesn't seem to make this easy. As a first step, I would gather the locations of where TreePlot planned on putting the vertices by running the function once an appending their names and locations onto a list to use:

Sow and Reap are often useful in these situations as well. "vlist" is now a list of pairs of "y" values and the corresponding vertex name.
Run Union on it to remove duplicate entries and to sort them:Gather the entries together by their first value:Use Map and Part together to strip out the y values:
This expression results in {{2, 3, 5, 7}, {4, 6, 8}, {1}}, which are the names of the vertices grouped together by their row from bottom to top. You can now use this in the construction of a VertexRenderingFunction for the original TreePlot.

Nice! Thanks very much!

I didn't see the issue at first myself, but if you insert some print statements, debugging shows that parent[[0]] gives the head, List. Mathematica indexes lists from 1 to their length instead of from 0 to their length-1, so replace "parent[[0]]" with "parent[[1]]"