Trees appear everywhere in higher-dimensional algebra. In this text they were defined in a purely abstract way (2.3.3): tr is the free plain operad on the terminal object of Setℕ, and an n-leafed tree is an element of tr(n). But for the reasons laid out at the beginning of Section 7.3, I give here a ‘concrete’, graph-theoretic, definition of (finite, rooted, planar) tree and sketch a proof that it is equivalent to the abstract definition.

The equivalence

The main subtlety is that the trees we use are not quite finite graphs in the usual sense: some of the edges have a vertex at only one of their ends. (Recall from 2.3.3 that in a tree, an edge with a free end is not the same thing as an edge ending in a vertex.) This suggests the following definitions. Definition E.1.1 A (planar) input-output graph (Fig. E-A(a)) consists of

• • a finite set V (the vertices)

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