  We analyze simple random walk on a supercritical Galton-Watson tree, where
the walk is conditioned to return to the root at time $2n$. Specifically, we
establish the asymptotic order (up to a constant factor) as $n\to\infty$, of
the maximal displacement from the root. Our results, which are shown to hold
for almost surely every surviving tree $T$ (subject to some mild moment
conditions), are broken up into two cases. When the offspring distribution
takes a value less than or equal to $1$ with positive probability, the maximal
displacement of the bridge is shown to be on the order of $n^{1/3}$.
Conversely, when the offspring distribution has minimum possible value equal to
at least $2$ (and is non-constant), the maximal displacement is shown to be of
order less than $n$, but greater than $n^{\gamma}$ (for any $\gamma<1$). Each
of these cases is in contrast to the case of a regular tree, on which the
bridge is known to be diffusive. To obtain our results, we show how the walk
tends to gravitate towards large clusters of vertices of minimal degree, where
it then proceeds to spend most of its time. The size and frequency of such
clusters is generally dependent on the minimum possible value attainable by the
offspring distribution, and it is this fact which largely accounts for the
existence of the two regimes.
