  In order to study the statistics of the objects with hierarchical merging, we
propose the skeleton tree formalism, which can analytically distinguish the
episodic merging and the continuous accretion in the mass growth processes. The
distinction was not clear in extended Press-Schechter (PS) formalism. The
skeleton tree formalism is a natural extension of the peak theory which is an
alternative formalism for the statistics of the bound objects. The fluctuation
field smoothing with Gaussian filter produces the landscape with adding the
extra-dimension of the filter resolution scale to the spatial coordinate of the
original fluctuation. In the landscape, some smoothing peaks are nesting into
the neighboring peaks at a type of critical points called sloping saddles
appears, which can be interpreted as merging events of the objects in the
context of the hierarchical structure formation. The topological properties of
the landscape can be abstracted in skeleton trees, which consist of line
process of the smoothing peaks and the point process of the sloping saddles.
According to this abstract topological picture, in this paper, we present the
concept and the basic results of the skeleton tree formalism to describe (1)
the distinction between the accretion and the merger in the hierarchical
structure formation from various initial random Gaussian fields; (2) the
instantaneous number density of the sloping saddles which gives the
instantaneous scale function of the objects with the destruction and
reformation in the mergers; (3) the rates of the destruction, the reformation,
and the relative accretion growth; (4) the self-consistency of the formalism
for the statistics of the mass growth processes of the objects; (5) the mean
growth history of the objects at the fixed mass.
