Plane tree

A plane tree is a tree embedded in an oriented plane, with the embedding considered up to deformation preserving the cyclic order of incident edges. The cyclic order distinguishes plane embeddings that an abstract graph isomorphism can forget.

After choosing a root, the parent edge gives a starting position for the children at each nonroot vertex. At the root there is no parent edge: a marked root corner is needed to turn its cyclic order into a linear child order. A rooted plane tree includes that extra choice, equivalently an ordered child list at every vertex. Merely distinguishing the root vertex does not select its first child.

A plane tree with a distinguished root and root corner, equivalently a rooted tree with a linear order on the children at each vertex. Trees with $n$ edges are counted by the Catalan number $C_n$.