  Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution
$\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails.
Consider the $\lambda$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$;
this is the nearest neighbor random walk which, when at a vertex $v$ with $d_v$
offspring, moves closer to the root with probability $\lambda/(\lambda+d_v)$,
and moves to each of the offspring with probability $1/(\lambda+d_v)$. It is
known that this walk has an a.s. constant speed $\v=\lim_n |X_n|/n$
  (where $|X_n|$ is the distance of $X_n$ from the root), with $\v>0$ for $
0<\lambda<m$ and $\v=0$ for $\lambda \ge m$. For all $\lambda \le m$, we prove
a quenched CLT for $|X_n|-n\v$. (For $\lambda>m$ the walk is positive
recurrent, and there is no CLT.) The most interesting case by far is
$\lambda=m$, where the CLT has the following form: for almost every ${\cal T}$,
the ratio $|X_{[nt]}|/\sqrt{n}$ converges in law as $n \to \infty$ to a
deterministic multiple of the absolute value of a Brownian motion. Our approach
to this case is based on an explicit description of an invariant measure for
the walk from the point of view of the particle (previously, such a measure was
explicitly known only for $\lambda=1$) and the construction of appropriate
harmonic coordinates.
