  We study bond percolation of the Cayley tree (CT) by focusing on the
probability distribution function (PDF) of a local variable, namely, the size
of the cluster including a selected vertex. Because the CT does not have a
dominant bulk region, which is free from the boundary effect, even in the
large-size limit, the phase of the system on it is not well defined. We herein
show that local observation is useful to define the phase of such a system in
association with the well-defined phase of the system on the Bethe lattice,
that is, an infinite regular tree without boundary. Above the percolation
threshold, the PDFs of the vertex at the center of the CT (the origin) and of
the vertices near the boundary of the CT (the leaves) have different forms,
which are also dissimilar to the PDF observed in the ordinary percolating phase
of a Euclidean lattice. The PDF for the origin of the CT is bimodal: a decaying
exponential function and a system-size-dependent asymmetric peak, which obeys a
finite-size-scaling law with a fractal exponent. These modes are respectively
related to the PDFs of the finite and infinite clusters in the nonuniqueness
phase of the Bethe lattice. On the other hand, the PDF for the leaf of the CT
is a decaying power function. This is similar to the PDF observed at a critical
point of a Euclidean lattice but is attributed to the nesting structure of the
CT around the boundary.
