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... .