  We study tree lengths in $\Lambda$-coalescents without a dust component from
a sample of $n$ individuals. For the total length of all branches and the total
length of all external branches we present laws of large numbers in full
generality. The other results treat regularly varying coalescents with exponent
1, which cover the Bolthausen-Sznitman coalescent. The theorems contain laws of
large numbers for the total length of all internal branches and of internal
branches of order $a$ (i.e. branches carrying $a$ individuals out of the
sample). These results transform immediately to sampling formulas in the
infinite sites model. In particular, we obtain the asymptotic site frequency
spectrum of the Bolthausen-Sznitman coalescent. The proofs rely on a new
technique to obtain laws of large numbers for certain functionals of decreasing
Markov chains.
