First-order theories of bounded trees

A maximal chain in a tree is called a path, and a tree is called bounded when all its paths contain leaves. This paper concerns itself with first-order theories of bounded trees. We establish some sufficient conditions for the existence of bounded end-extensions that are also partial elementary extensions of a given tree. As an application of tree boundedness, we obtain a conditional axiomatisation of the first-order theory of the class of trees whose paths are all isomorphic to some ordinal α<ωω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha < \omega ^{\omega }$$\end{document}, given the first-order theories of certain classes of bounded trees.