For α less than ε0 let Nα be the number of occurrences of ω in the Cantor normal form of α. Further let |n | denote the binary length of a natural number n, let |n|h denote the h-times iterated binary length of n and let inv(n) be the least h such that |n|h ≤ 2. 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 〈α0,..., αn 〉 of ordinals less than ε0 which satisfy the condition that the Norm Nαi of the i-th term αi is bounded by K + |i | · |i|h. 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 〈α0,..., αn 〉 of ordinals less than ε0 which satisfies the condition that the Norm Nαi of the i-th term αi is bounded by K +|i|· 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 Matouˇsek and Loebl we obtain similar characterizations for finite bad sequences of finite trees in terms of Otter’s tree constant 2.9557652856.... ∗ Research supported by a Heisenberg-Fellowship of the Deutsche Forschungsgemeinschaft. † The main results of this paper were obtained during the authors visit of T. Arai in

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