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The small Veblen ordinal is the limit of the following sequence of ordinals $$\varphi(1, 0),\varphi(1, 0, 0),\varphi(1, 0, 0, 0),\varphi(1, 0, 0, 0, 0),\ldots$$ where $$\varphi$$ is the Veblen hierarchy as extended to arbitrary finite numbers of arguments. Using Buchholz's function it can be expressed as $$\psi_0(\psi_1(\psi_1(\psi_1(1))))$$, and using Weiermann's $$\vartheta$$ function, it can be expressed as $$\vartheta(\Omega^\omega)$$.

Harvey Friedman's tree(n) function (for unlabeled trees) is believed to grow at around the same rate as $$f_{\vartheta(\Omega^\omega)}(n)$$ in the fast-growing hierarchy with respect to an unspecified system of fundamental sequences.[citation needed]