# edge appearance probability and conditional independence

So I'm doing research on graphical models and on page 362 of http://www.seas.upenn.edu/~taskar/pubs/aistats09.pdf, it says that

"if $\beta_{uv}=0$ (i.e. weight of edge $uv$ is zero), edge $e_{uv}$ appears in trees with zero probability (though this does not mean $u$ and $v$ are conditionally independent in the ET model)."

Does anyone have an insight about why this could be true? The ET model (ensemble-of-trees model) is a type of graphical model that considers all possible spanning trees, as described in the paper. And in a graphical model, one would expect that the absence of an edge (probability of appearing is 0) is equivalent to conditional independence between the two nodes...

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