  We study the distribution of prime chains, which are sequences p_1,...,p_k of
primes for which p_{j+1}\equiv 1\pmod{p_j} for each j. We give estimates for
the number of chains with p_k\le x (k variable), and the number of chains with
p_1=p and p_k \le px. The majority of the paper concerns the distribution of
H(p), the length of the longest chain with p_k=p, which is also the height of
the Pratt tree for p. We show H(p)\ge c\log\log p and H(p)\le (\log p)^{1-c'}
for almost all p, with c,c' explicit positive constants. We can take, for any
\epsilon>0, c=e-\epsilon assuming the Elliott-Halberstam conjecture. A
stochastic model of the Pratt tree, based on a branching random walk, is
introduced and analyzed. The model suggests that for most p, H(p) stays very
close to e \log\log p.

Number Theory;Probability