If
x
x
is pendant in
G
G
, then
x
∗
{x^ \ast }
denotes the unique vertex of
G
G
adjacent to
x
x
. Such an
x
x
is said to be neighborhood-fixed whenever
x
∗
{x^ \ast }
is fixed by
A
(
G
−
x
)
A(G - x)
. It is shown that if
G
G
is not a tree and has a pendant vertex, but no *-fixed pendant vertex, then there is a subgraph
G
#
{G^\# }
of
G
G
such that for some
y
∈
V
(
G
#
)
y \in V({G^\# })
,
O
(
A
(
G
#
)
y
)
⩾
t
!
O(A{({G^\# })_y}) \geqslant t!
where
t
t
is the maximum number of edges in a tree rooted in
G
#
{G^\# }
.