What does "a universal tree" mean?

It is one of the concepts used in "ON THE REPRESENTATION OF CONTINUOUS FUNCTIONS OF SEVERAL VARIABLES AS SUPERPOSITIONS OF CONTINUOUS FUNCTIONS OF A SMALLER NUMBER OF VARIABLES", in the second paragraph, highlighted with black background, as shown in this image.

• Does this answer your question? What is a universal tree? Commented Jan 15, 2023 at 6:38
• @RyanBudney, the OP is asking about $\mathbb{R}$-trees, which are a different kind of trees. Commented Jan 15, 2023 at 12:22
• That's a little sad. Apparently "universal trees" are not universal. Commented Jan 15, 2023 at 20:13

In the beginning of Section 6 on p. 318 of Menger's Kurventheorie, referred to in Kolmogorov's paper on the page whose image you linked in your post, we find this:

Wir bezeichen eine Baumkurve $$B$$ als einen Universalbaum bzw. als Universalbaum $$n$$-ter Ordnung, wenn $$B$$ zu jedem vorgelegten Baum bzw. zu jedem vorgelegten Baum $$n$$-ter Ordnung eine homöomorphe Menge als Teilmenge enthält.

which apparently means

We refer to a tree curve $$B$$ as a universal tree, or a universal tree of the $$n$$th order, if $$B$$ contains as a subset a homeomorphic image of any given tree, or any given tree of the $$n$$th order.

• I think you're right,but when i read the passage,it's too hard to find the electronic edition of second quoted passage ,where did you find it? Commented Jan 16, 2023 at 14:56
• @mmmooo : I quoted only one passage. The second highlighted piece of text is just my attempt at translation (aided by Google Translate) of the first highlighted piece of text (in German, which I don't know). Commented Jan 16, 2023 at 15:03
• I mean to say,where did you find the k. menger's book kurventheorie,which is the second quotation of the Kolmogorov's paper.😂 Commented Jan 16, 2023 at 16:27