  Merger trees follow the growth and merger of dark-matter haloes over cosmic
history. As well as giving important insights into the growth of cosmic
structure in their own right, they provide an essential backbone to
semi-analytic models of galaxy formation. This paper is the first in a series
to arise from the SUSSING MERGER TREES Workshop in which ten different
tree-building algorithms were applied to the same set of halo catalogues and
their results compared. Although many of these codes were similar in nature,
all algorithms produced distinct results. Our main conclusions are that a
useful merger-tree code should possess the following features: (i) the use of
particle IDs to match haloes between snapshots; (ii) the ability to skip at
least one, and preferably more, snapshots in order to recover subhaloes that
are temporarily lost during merging; (iii) the ability to cope with (and
ideally smooth out) large, temporary flucuations in halo mass. Finally, to
enable different groups to communicate effectively, we defined a common
terminology that we used when discussing merger trees and we encourage others
to adopt the same language. We also specified a minimal output format to record
the results.
