For \al less than \eo let N\al be the number of occurrences
of \om in the Cantor normal form of \al. Further let \lh n
denote the binary length of a natural number n, let \lhh n denote
the h-times iterated binary length of n and let \inv n be the
least h such that \lhh n \leq2. We show that for any natural number
h first order Peano arithmetic, \PA, does not prove the following
sentence: For all K there exists an M which bounds the lengths n
of all strictly descending sequences $\langle \al0,... ,\aln\rangle
of ordinals less than \eo which satisfy the condition that the
Norm N\ali of the i-th term \ali is bounded by K + \lh i\cdot \lhh{i}.
¶ As a supplement to this (refined Friedman style) independence result we further
show that e.g., primitive recursive arithmetic, \PRA, proves
that for all K there is an M which bounds the length n of any strictly
descending sequence \langle \al0,... ,\aln\rangle of ordinals less than
\eo which satisfies the condition that the Norm N\ali of the i-th term
\ali is bounded by K + \lh i \cdot \inv i$. The proofs are based on results
from proof theory and techniques from asymptotic analysis of Polya-style enumerations.
¶ Using results from Otter and from Matou{\v{s}}ek and Loebl we obtain similar
characterizations for finite bad sequences of finite trees in terms of Otter's
tree constant 2.9557652856... .