  We study properties of the harmonic measure of balls in typical large
discrete trees. For a ball of radius $n$ centered at the root, we prove that,
although the size of the boundary is of order $n$, most of the harmonic measure
is supported on a boundary set of size approximately equal to $n^{\beta}$,
where $\beta\approx0.78$ is a universal constant. To derive such results, we
interpret harmonic measure as the exit distribution of the ball by simple
random walk on the tree, and we first deal with the case of critical
Galton-Watson trees conditioned to have height greater than $n$. An important
ingredient of our approach is the analogous continuous model (related to
Aldous' continuum random tree), where the dimension of harmonic measure of a
level set of the tree is equal to $\beta$, whereas the dimension of the level
set itself is equal to $1$. The constant $\beta$ is expressed in terms of the
asymptotic distribution of the conductance of large critical Galton-Watson
trees.
