  In this article we compare the known dynamical polynomial time algorithm for
the game-over attack strategy, to that of the brute force approach; of checking
all the ordered rooted subtrees of a given tree that represents a given
computer network. Our approach is purely enumerative and combinatorial in
nature. We first revisit known results about a doubly exponential sequence and
generalize them. We then consider both finite and locally finite ordered rooted
trees (LFOR-trees), and the class of their finite ordered rooted subtrees of
bounded height, describing completely the LFOR-trees with no leaves where the
number of ordered rooted subtrees of height at most $h$ are bounded by a
polynomial in $h$. We finally consider general LFOR-trees where each level can
have leaves and determine conditions for the number of ordered rooted subtrees
of height at most $h$ to be bounded by a polynomial in $h$.
