This paper deals with
characterizing two types cf monotone, upper semi-continuous decompositions of a
Piausdoffi continuum that is irreducible about a finite subset. One of the
decompositions is minimal with respect lo the property of having a quotient space
which is a tree (a hereditarily unicoherent, locally connected continuum) and is
characterized in terms of certain collections of subcontinua. The other decomposition
is not only minimal but also unique with respect to the properties that the
quotient space is a tree and the elements of the decomposition have void
interiors. This decomposition is characterized quite simply by prohibiting
the existence of indecomposable subcontinua with nonvoid interiors. The
stmcture of the elements of Che decompositions that have void interiors is very
nice and is described by means of the aposyndetic se5 function T. In the
case where elements exist with nonvoid interiors, the stmcture can be very
complicated and a final result deals with this structure under some rather stringent
conditions.
