Synchronize the profiler state with the document

This is suboptimal, because mutation leaves room for subtle bugs, but
rewriting @tebbi's code to be functional was a pain, and not something I
could figure out how to do easily.  I'm working under the assumption
that there is no sharing in a single treenode, which I'm not completely
sure is valid.  That said, a few simple tests seem to indicate that this
works as expected.

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