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We study tree lengths in $\Lambda$-coalescents without a dust component froma sample of $n$ individuals. For the total length of all branches and the totallength of all external branches we present laws of large numbers in fullgenerality. The other results treat regularly varying coalescents with exponent1, which cover the Bolthausen-Sznitman coalescent. The theorems contain laws oflarge numbers for the total length of all internal branches and of internalbranches of order $a$ (i.e. branches carrying $a$ individuals out of thesample). These results transform immediately to sampling formulas in theinfinite sites model. In particular, we obtain the asymptotic site frequencyspectrum of the Bolthausen-Sznitman coalescent. The proofs rely on a newtechnique to obtain laws of large numbers for certain functionals of decreasingMarkov chains.

Tree lengths for general $\Lambda$-coalescents and the asymptotic site frequency spectrum around the Bolthausen-Sznitman coalescent