uniform spanning tree of $2 \times n$ graph

Question

In Probability on Trees and Networks Chapter 1 study the uniform spanning tree on the ladder graph:

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The probability the bottom rung appears in a uniform spanning tree of this graph is $\sqrt{3}-1$ and for finite ladders, the probabilities are continued fraction convergents.

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Why is the limit $\sqrt{3}-1$ and why do continued fractions appear here?

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Here is a uniform spanning tree of a ladder graph and indeed, you get the bottom rung 7 in 10 times.

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Maybe the identity $\frac{1}{\sqrt{3}-1} = \frac{\sqrt{3}+1}{2}$ may be useful.