Hausdorff dimension of harmonic measure for Galton Watson trees The Holder exponent of a probability measure $\mu$ at $x$, in general, is defined to be:
$$H\ddot{o}(\mu)(x) = \lim_{r\downarrow 0} \frac{\log\left( \frac{1}{\mu(B_r(x))}\right)}{\log\left( \frac{1}{r}\right)} \tag{1}$$
For Galton Watson trees $T$, we define $\partial T$, the boundary of $T$, to be the set of all rays emanating from the root of the tree. There is a result that for any Borel probability measure $\mu$ on the boundary of a tree $T$, if the Holder exponent exists and is a constant, then that constant is the Hausdorff dimension of $\mu$.
So this paper (page $11$), says that for a Borel probability measure $\theta$ on $\partial T$:
$$H\ddot{o}(\theta)(\xi) = \lim_{n\to \infty}\frac{1}{n}\log\left(\frac{1}{\theta(\xi_n)} \right) \tag{2}$$
where $\xi\in \partial T$ and $\xi_n$ is the $n^{th}$ vertex along $\xi$. Notation is defined in detail in the paper above in page $5$. 

I don't understand how definitions of $(1)$ and $(2)$ are equivalent i.e. $B_r(x)$ is a neighborhood around point $x$ and so I was expecting a neighborhood around $\xi$ in $(2)$ which I don't know how it looks like and how that leads to the definition of $(2)$?

 A: They use curious notions and notations (people familiar with symbolic dynamics surely will notice that they are rewriting the common notions differently). In particular, the usual name for "Hölder exponent of a probability measure" is "pointwise dimension of a probability measure".
Letting aside these aspects, with their definition
$$
\theta(x) = \mu (\{\xi \in \partial T : x \in \xi\}),
$$
the translation of $\theta(\xi_n)$ is: the measure of elements of the boundary starting from the $n$th vertex of $\xi$. The trick is then to take $r=e^{-n+1}$ (since the limit exists, although it would still be ok with limsup and liminf) and use their distance $d$ (see page 6 of the paper) to understand that for this value of $r$ we have
$$
B_r(\xi)=\{\eta\in \partial T:e^{-|\eta\wedge \xi|}<e^{-n+1}\}=\{\eta \in \partial T : \xi_n \in \eta\}.
$$

A minor improvement of what you wrote that sometimes is crucial:
For any Borel probability measure $\mu$ on the boundary of a tree, if the Hölder exponent exists and is a constant almost everywhere, then that constant is the Hausdorff dimension of $\mu$.

