 Solution

The distinguished chain generated by zero is countable, and every other component in the classification of models of the successor theory is a countable copy of the integer successor chain. A model of cardinality ${\aleph }_1$ therefore has exactly ${\aleph }_1$ additional components: a finite or countable collection would be countable, while the union of $\kappa$ countable components has size $\max (\kappa ,{\aleph }_0)$.

Given two such models, match their zero chains in the forced way and choose a bijection between their additional components. On each component choose an origin and match corresponding integer iterates; this commutes with successor. The union of these maps is an isomorphism of structures. Existence is given by a disjoint union with ${\aleph }_1$ integer chains. HenceThe same argument works at every uncountable cardinal. Countable models can still differ by the number of extra components, as part (ii) shows.