Let
G
=
Π
∗
(
A
i
;
U
j
k
=
U
k
j
)
G = {\Pi ^ \ast }({A_i};{U_{jk}} = {U_{kj}})
be a tree product with
H
H
a subgroup of
G
G
. By extending the technique of using a rewriting process we show that
H
H
is an HNN group whose base is a tree product with vertices of the form
x
A
i
x
−
1
∩
H
x{A_i}{x^{ - 1}} \cap H
. The associated subgroups are contained in vertices of the base, and both the associated subgroups of
H
H
and the edges of its base are of the form
y
U
j
k
y
−
1
∩
H
y{U_{jk}}{y^{ - 1}} \cap H
. The
x
x
and
y
y
are certain double coset representatives for
G
mod
(
H
,
A
i
)
G\bmod (H,{A_i})
and
G
mod
(
H
,
U
j
k
)
G\bmod (H,{U_{jk}})
, respectively, and the elements defined by the free part of
H
H
are specified. More precise information about
H
H
is given when
H
H
is either indecomposable or
H
H
satisfies a nontrivial law. Introducing direct tree products, we use our subgroup theorem to prove that if each edge of
G
G
is contained in the center of its two vertices then the cartesian subgoup of
G
G
is a free group. We also use our subgroup theorem in proving that if each edge of
G
G
is a finitely generated subgroup of finite index in both of its vertices and some edge is a proper subgroup of both its vertices then
G
G
is a finite extension of a free group iff the orders of the
A
i
{A_i}
are uniformly bounded.