  We study the number of random records in an arbitrary split tree (or
equivalently, the number of random cuttings required to eliminate the tree). We
show that a classical limit theorem for convergence of sums of triangular
arrays to infinitely divisible distributions can be used to determine the
distribution of this number. After normalization the distributions are shown to
be asymptotically weakly 1-stable. This work is a generalization of our earlier
results for the random binary search tree, which is one specific case of split
trees. Other important examples of split trees include $m$-ary search trees,
quadtrees, medians of $(2k+1)$-trees, simplex trees, tries and digital search
trees.
